From c71dec00b4ccb5c3a8ff3939e040c9d9304c682d Mon Sep 17 00:00:00 2001 From: cyschneck Date: Mon, 23 Dec 2024 22:36:12 +0000 Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20ProjectP?= =?UTF-8?q?ythia/great-circle-cookbook@3d4fab81ea88c81d4175a96b05a677807ea?= =?UTF-8?q?a0118=20=F0=9F=9A=80?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../notebooks/foundations/1_terminology.ipynb | 36 ++++++--------- .../notebooks/foundations/2_coordinates.ipynb | 24 +++++----- .../notebooks/tutorials/1_arc_path.ipynb | 42 ++++++++--------- .../notebooks/tutorials/2_arc_to_point.ipynb | 12 ++--- .../tutorials/3_parallels_max_min.ipynb | 38 +++++---------- .../tutorials/4_path_intersection.ipynb | 15 ++---- .../notebooks/tutorials/5_angles.ipynb | 8 ++-- .../notebooks/tutorials/6_polygon_area.ipynb | 13 ++---- .../notebooks/foundations/1_terminology.html | 46 ++++++++----------- .../notebooks/foundations/2_coordinates.html | 23 ++++------ _preview/1/notebooks/notebook-template.html | 4 +- .../1/notebooks/tutorials/1_arc_path.html | 40 ++++++++-------- .../1/notebooks/tutorials/2_arc_to_point.html | 10 ++-- .../tutorials/3_parallels_max_min.html | 26 +++++------ .../tutorials/4_path_intersection.html | 6 +-- _preview/1/notebooks/tutorials/5_angles.html | 6 +-- .../1/notebooks/tutorials/6_polygon_area.html | 4 +- _preview/1/searchindex.js | 2 +- 18 files changed, 153 insertions(+), 202 deletions(-) diff --git a/_preview/1/_sources/notebooks/foundations/1_terminology.ipynb b/_preview/1/_sources/notebooks/foundations/1_terminology.ipynb index f899692..35cc829 100644 --- a/_preview/1/_sources/notebooks/foundations/1_terminology.ipynb +++ b/_preview/1/_sources/notebooks/foundations/1_terminology.ipynb @@ -10,7 +10,7 @@ "tags": [] }, "source": [ - "\"https://www.jpl.nasa.gov/edu/news/2022/12/8/nasa-mission-takes-a-deep-dive-into-earths-surface-water/" + "\"https://www.jpl.nasa.gov/edu/news/2022/12/8/nasa-mission-takes-a-deep-dive-into-earths-surface-water/" ] }, { @@ -36,7 +36,7 @@ "\n", "While spherical geometry played an important role historically in the fields of astronomy and navigation, its teaching has largely fallen out of favor since the 1950's making finding comphrenshive resources difficult.\n", "\n", - "This notebook will cover some of the important and unique terminology used when working with Great Circles and spherical geometry\n", + "This notebook will cover some of the important and unique terminology used when working with Great Circles and spherical geometry.\n", "\n", "1. Spherical Geometry\n", "1. Great Circles\n", @@ -77,12 +77,6 @@ "\n", "\"Great\n", "\n", - "### Spherical Triangles\n", - "TODO: via Spherical trigonometry (Fajardo)\n", - "\n", - "### Napier's Rules\n", - "TODO: via Spherical trigonometry (Fajardo)\n", - "\n", "### Law of Cosines\n", "> \"The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule\" [(Wikiepedia)](https://en.wikipedia.org/wiki/Spherical_trigonometry)\n", "\n", @@ -93,7 +87,7 @@ "$$cos(c) = cos(a)cos(b) + sin(a)sin(b)cos(C)$$\n", "### Law of Sines\n", "\n", - "The spherical law of sines states that angles A, B, and C be the angles opposite of the sides a, b, c, where\n", + "The spherical law of sines states that angles A, B, and C be the angles opposite of the sides a, b, c, where:\n", "\n", "$$\\frac{sin(A)}{sin(a)} = \\frac{sin(B)}{sin(b)} = \\frac{sin(C)}{sin(c)}$$\n", "\n", @@ -108,7 +102,7 @@ "\n", "### Great Circle Path vs. Great Circle Arc\n", "\n", - "A great circle is the largest circle that can be formed on the surface of a sphere created by the intersection of a plane that also passes through the center of the sphere\n", + "A great circle is the largest circle that can be formed on the surface of a sphere created by the intersection of a plane that also passes through the center of the sphere.\n", "\n", "All Great Circles:\n", "- Intersect the center of the Earth\n", @@ -128,9 +122,7 @@ "source": [ "## Ellipsoids vs. Spheres\n", "\n", - "The earth is not round, instead it is an irregular ellipsoid known as a an oblate spheroid where the poles are slightly flatter. Spherical trigonomeyry assumes a unit sphere, so working on Earth, without additional corrections, spherical measurements will contain up to 0.3% (22 km) when assuming the Earth is a perfect sphere\n", - "\n", - "> TODO: https://gis.stackexchange.com/questions/25494/how-accurate-is-approximating-earth-as-sphere\n", + "The earth is not round, instead it is an irregular ellipsoid known as a an oblate spheroid where the poles are slightly flatter. Spherical trigonomeyry assumes a unit sphere, so working on Earth, without additional corrections, spherical measurements will contain up to 0.3% (22 km) when assuming the Earth is a perfect sphere ([see more](https://gis.stackexchange.com/questions/25494/how-accurate-is-approximating-earth-as-sphere)).\n", "\n", "To account for the error when assuming the Earth is a sphere, there are various geodetic systems and ellipsoids to include in calculations." ] @@ -146,7 +138,7 @@ "- [`pyproj`: Python interface to PROJ (catographic projections and coordinate transformations library)](https://pyproj4.github.io/pyproj/stable/)\n", "- [`geopy`: Python client for several popular geocoding web services](https://geopy.readthedocs.io/en/stable/#)\n", "\n", - "`pyproj` and `geopy` both take advantage of different types of (optional) ellipsoids" + "`pyproj` and `geopy` both take advantage of different types of (optional) ellipsoids." ] }, { @@ -241,11 +233,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "**The standard reference ellipsoid for working with Earth is WGS-84**\n", + "### **The standard reference ellipsoid for working with Earth is WGS-84**\n", "\n", - "`geopy` by default makes use of WGS-84 where \"the mean earth radius as defined by the International Union of Geodesy and Geophysics, (2a + b)/3 = 6371.0087714150598 kilometers approx 6371.009 km (for WGS-84), resulting in an error of up to about 0.5%\" ([geopy](https://geopy.readthedocs.io/en/stable/))\n", + "`geopy` by default makes use of WGS-84 where \"the mean earth radius as defined by the International Union of Geodesy and Geophysics, (2a + b)/3 = 6371.0087714150598 kilometers approx 6371.009 km (for WGS-84), resulting in an error of up to about 0.5%\" ([geopy](https://geopy.readthedocs.io/en/stable/)).\n", "\n", - "WGS-84 is a unified global ellipsoid model that is used for GPS collected from GPS satellites to calculate extremely preise measurements of the Earth. For the purpose of this notebook, this is the ellipsoid model we will be working with" + "WGS-84 is a unified global ellipsoid model that is used for GPS collected from GPS satellites to calculate extremely preise measurements of the Earth. For the purpose of this notebook, this is the ellipsoid model we will be working with." ] }, { @@ -271,7 +263,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "WGS-84 is an ellipsoid with a semi-major axis of `6378137.0` meters, an inverse flattening feature of `298.257223563`, and a flattening factor of `0.0033528106647474805`\n", + "WGS-84 is an ellipsoid with a semi-major axis of `6378137.0` meters, an inverse flattening feature of `298.257223563`, and a flattening factor of `0.0033528106647474805`.\n", "\n", "[Learn more!](https://gisgeography.com/wgs84-world-geodetic-system/)" ] @@ -282,7 +274,7 @@ "source": [ "## Geodesy\n", "\n", - "Geodesy is the complex science of measuring the Earth's \"geometric shape, orientation in space, and gravity field\"\n", + "Geodesy is the complex science of measuring the Earth's \"geometric shape, orientation in space, and gravity field\".\n", "\n", "[Learn more!](https://oceanservice.noaa.gov/facts/geodesy.html)\n", "\n", @@ -331,10 +323,10 @@ "source": [ "## Summary\n", "\n", - "A great circle is formed by a plane intersecting a sphere and the center, like the equator\n", + "A great circle is formed by a plane intersecting a sphere and the center, like the equator.\n", "\n", "\n", - "Great Circles make use of spherical geometry to measure features on the curved surface of a unit sphere. However, planets like Earth are not perfect spheres and to account for the error are combined with geodesic calculations to reduce the error in final calculations\n", + "Great Circles make use of spherical geometry to measure features on the curved surface of a unit sphere. However, planets like Earth are not perfect spheres and to account for the error are combined with geodesic calculations to reduce the error in final calculations.\n", "\n", "### What's next?\n", "\n", @@ -358,7 +350,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.12.7" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/_sources/notebooks/foundations/2_coordinates.ipynb b/_preview/1/_sources/notebooks/foundations/2_coordinates.ipynb index 6c688d7..07b0d19 100644 --- a/_preview/1/_sources/notebooks/foundations/2_coordinates.ipynb +++ b/_preview/1/_sources/notebooks/foundations/2_coordinates.ipynb @@ -32,7 +32,7 @@ "metadata": {}, "source": [ "## Overview\n", - "Understanding different types of coordinates for working with unit spheres and ellipsoids\n", + "Understanding different types of coordinates for working with unit spheres and ellipsoids.\n", "\n", "1. Types of Coordinates\n", "1. Convert Coordinates to All Coordinate Types\n", @@ -94,9 +94,7 @@ "source": [ "### Geodesic Coordinates\n", "\n", - "> A geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude [(Wikipedia)](https://en.wikipedia.org/wiki/Geographic_coordinate_system)\n", - "\n", - "Geodesic coordinates are latitude and longtiude, from -90° South to 90° North and -180° East to 180° West measured from Greenwich\n", + "Geodesic coordinates are latitude and longtiude, from -90° South to 90° North and -180° East to 180° West measured from Greenwich.\n", "\n", "

\n", " \"Longitude\n", @@ -109,7 +107,7 @@ "source": [ "### Cartesian Coordinates\n", "\n", - "Cartesian coordinates describe points in space based on perpendicular axis lines that meet at a singlle point of origin, where any point's position is described based on the distance to the origin along xyz axis\n", + "Cartesian coordinates describe points in space based on perpendicular axis lines that meet at a single point of origin, where any point's position is described based on the distance to the origin along xyz axis.\n", "\n", "

\n", " \"A\n", @@ -119,7 +117,7 @@ "\n", "**Geodesic to Cartesian Coordinates**\n", "\n", - "Assuming the earth's radius is 6378137 meters\n", + "Assuming the earth's radius is 6378137 meters:\n", "\n", "$$x = radius * cos(latitude) * cos(longitude)$$\n", "$$y = radius * cos(latitude) * sin(longitude)$$\n", @@ -148,7 +146,7 @@ "source": [ "### Spherical Coordinates\n", "\n", - "Spherical coordinates describe points in space based on three values: radial distance (rho, r) along the radial line between point and the origin, polar angle (theta, θ) between the radial line and the polar axis, and azimuth angle (phi, φ) which is the angle of rotation of the radial line around the polar axis. With a fixed radius, the 3-point coordinates (r, θ, φ) provide a coordinate along a sphere\n", + "Spherical coordinates describe points in space based on three values: radial distance (rho, r) along the radial line between point and the origin, polar angle (theta, θ) between the radial line and the polar axis, and azimuth angle (phi, φ) which is the angle of rotation of the radial line around the polar axis. With a fixed radius, the 3-point coordinates (r, θ, φ) provide a coordinate along a sphere.\n", "\n", "- Radial distance: distance from center to surface of sphere\n", "- Polar angle: angle between radial line and polar axis\n", @@ -192,9 +190,9 @@ "source": [ "### Polar Coordinates\n", "\n", - "Polar coordinates are a combination of latitude, longitude, and altitude from the center of the sphere (based on the radius)\n", + "Polar coordinates are a combination of latitude, longitude, and altitude from the center of the sphere (based on the radius).\n", "\n", - "Assuming the earth's radius is 6378137 meters\n", + "Assuming the earth's radius is 6378137 meters:\n", "\n", "$$x = cos(latitude) * cos(longitude) * radius$$\n", "$$y = cos(latitude) * sin(longitude) * radius$$\n", @@ -230,7 +228,7 @@ "source": [ "### Display Coordinates of Cities\n", "\n", - "Read in latitude and longitude coordinates from locations" + "First we will read in the latitude and longitude coordinates from locations:" ] }, { @@ -1026,9 +1024,9 @@ "source": [ "## Summary\n", "\n", - "Coordinates on the Earth are measured in many different types of coordinate systems: Geodesic (latitude/longitude), cartesian, spherical, and polar. These coordinates will make future calculations simpler by converting a 2D coordinate like latitude/longitude into a 3D space that can be used for vector calculations\n", + "Coordinates on the Earth are measured in many different types of coordinate systems: Geodesic (latitude/longitude), cartesian, spherical, and polar. These coordinates will make future calculations simpler by converting a 2D coordinate like latitude/longitude into a 3D space that can be used for vector calculations.\n", "\n", - "In Python, coordinates can be mapped on to a world map via Matploblib and Cartopy\n", + "In Python, coordinates can be mapped on to a world map via `matplotlib` and `cartopy`.\n", "\n", "### What's next?\n", "\n", @@ -1067,7 +1065,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.8" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/_sources/notebooks/tutorials/1_arc_path.ipynb b/_preview/1/_sources/notebooks/tutorials/1_arc_path.ipynb index acae119..2d18763 100644 --- a/_preview/1/_sources/notebooks/tutorials/1_arc_path.ipynb +++ b/_preview/1/_sources/notebooks/tutorials/1_arc_path.ipynb @@ -27,9 +27,9 @@ "source": [ "## Overview\n", "\n", - "Imagine a plane, flying from Cario to Hong Kong. To a passenger, the plane appears to travel a straight path, but the plane actually curves around the surface, held down by the gravity of the planet\n", + "Imagine a plane flying from Cario to Hong Kong. To a passenger, the plane appears to travel a straight path, but the plane actually curves around the surface, held down by the gravity of the planet.\n", "\n", - "Great circles are circles that circumnavigate the globe\n", + "Great circles are circles that circumnavigate the globe.\n", "\n", "- Distance between Points on a Great Circle Arc\n", "- Spherical Distance to Degrees\n", @@ -257,11 +257,11 @@ "source": [ "### Determine Distance Mathematically via Unit Sphere\n", "\n", - "Distance between point A (latA, lonA) and point B (latB, lonB)\n", + "Distance between point A (latA, lonA) and point B (latB, lonB):\n", "\n", "$$d=acos(sin(latA)*sin(latB)+cos(latA)*cos(latB)*cos(lonA-lonB))$$\n", "\n", - "For shorter distances (with less rounding errors)\n", + "For shorter distances (with less rounding errors):\n", "\n", "$$d=2*asin(\\sqrt{sin(\\frac{latA-latB}{2})^2 + cos(latA)*cos(latB)*sin(\\frac{lonA-lonB}{2})^2})$$\n", "\n", @@ -306,11 +306,11 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Additional distance measuerments\n", - "- Haversine (TODO)\n", - "- Vincenty Sphere Great Circle Distance (TODO)\n", - "- Vincenty Ellipsoid Great Circle Distance (TODO)\n", - "- Meeus Great Circle Distance (TODO)" + "Additional types of distance measuerments:\n", + "- Haversine\n", + "- Vincenty Sphere Great Circle Distance\n", + "- Vincenty Ellipsoid Great Circle Distance\n", + "- Meeus Great Circle Distance" ] }, { @@ -319,7 +319,7 @@ "source": [ "### Determine Distance Points via Python Package `pyproj`\n", "\n", - "`pyproj` accounts for different ellipsoids like `WGS-84`\n", + "`pyproj` accounts for different ellipsoids like `WGS-84`.\n", "\n", "First, setup a ellipsoid to represent the Earth (\"WGS-84\"):" ] @@ -358,7 +358,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Compared to the distance from the associated airports in Denver and Boston ([DIA to Logan](https://www.greatcirclemap.com/?routes=DEN-BOS)) which has a distance of 2823 km, using Denver instead of Boulder" + "Compared to the distance from the associated airports in Denver and Boston ([DIA to Logan](https://www.greatcirclemap.com/?routes=DEN-BOS)) which has a distance of 2823 km, using Denver instead of Boulder." ] }, { @@ -367,7 +367,7 @@ "source": [ "## Spherical Distance to Degrees\n", "\n", - "Convert a distance from meters to degrees, measured along the great circle sphere with a constant spherical radius of ~6371 km (mean radius of Earth)\n", + "Convert a distance from meters to degrees, measured along the great circle sphere with a constant spherical radius of ~6371 km (mean radius of Earth).\n", "\n", "- See also: [ObsPy `kilometer2degrees()`](https://docs.obspy.org/packages/autogen/obspy.geodetics.base.kilometer2degrees.html)" ] @@ -420,7 +420,7 @@ "source": [ "### Determine the Bearing Mathematically via Unit Sphere\n", "\n", - "Bearing between point A (latA, lonA) and point B (latB, lonB)\n", + "Bearing between point A (latA, lonA) and point B (latB, lonB):\n", "\n", "$$x = cos(latA) * sin(latB) - sin(latA) * cos(latB) * cos(lonB - lonA)$$\n", "$$y = sin(lonB - lonA) * cos(latB)$$\n", @@ -453,7 +453,7 @@ "source": [ "### Determine the Bearing via Python Package `pyproj`\n", "\n", - "`pyproj` accounts for different ellipsoids like `WGS-84`" + "`pyproj` accounts for different ellipsoids like `WGS-84`:" ] }, { @@ -505,7 +505,7 @@ "metadata": {}, "source": [ "### Determine Intermediate Points Mathemetically via Unit Sphere\n", - "Determine the points (lat, lon) a given fraction of a distance (d) between a starting points A (latA, lonA) and the final point B (latB, lonB) where f is a fraction along the great circle arc. `f=0` is point A and `f=1` is point B\n", + "Determine the points (lat, lon) a given fraction of a distance (d) between a starting points A (latA, lonA) and the final point B (latB, lonB) where f is a fraction along the great circle arc. `f=0` is point A and `f=1` is point B.\n", "\n", "> Note: The points cannot be antipodal because the path is undefined\n", "\n", @@ -1059,7 +1059,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The midpoint of an arc can be determiend as a fractional distance along an arc where f = 0.5" + "The midpoint of an arc can be determiend as a fractional distance along an arc where f = 0.5." ] }, { @@ -1142,7 +1142,7 @@ "source": [ "## Generate a Great Circle Path\n", "\n", - "Get points on the Great Cricle defined by two points" + "Get points on the Great Cricle defined by two points." ] }, { @@ -1341,7 +1341,7 @@ "source": [ "## Determine an Antipodal Point\n", "\n", - "Antipodal is the point on the globe that is on the exact opposite side of the Earth\n", + "Antipodal is the point on the globe that is on the exact opposite side of the Earth.\n", "\n", "Antipodal latitude is defined as:\n", "\n", @@ -1516,11 +1516,11 @@ "source": [ "## Summary\n", "\n", - "Calculating and mapping the midpoint and intermediate points along the great circle arc and closed circle path\n", + "Calculating and mapping the midpoint and intermediate points along the great circle arc and closed circle path.\n", "\n", "### What's next?\n", "\n", - "With a great circle arc defined, determine if a third point is along the arc or at what distance it sits from the great circle arc and path\n" + "With a great circle arc defined, determine if a third point is along the arc or at what distance it sits from the great circle arc and path.\n" ] }, { @@ -1555,7 +1555,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.8" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/_sources/notebooks/tutorials/2_arc_to_point.ipynb b/_preview/1/_sources/notebooks/tutorials/2_arc_to_point.ipynb index c71403c..f0b4aed 100644 --- a/_preview/1/_sources/notebooks/tutorials/2_arc_to_point.ipynb +++ b/_preview/1/_sources/notebooks/tutorials/2_arc_to_point.ipynb @@ -26,7 +26,7 @@ "metadata": {}, "source": [ "## Overview\n", - "A plane traveling across the country suddenly discovers it is low on fuel! It can no longer make it to the distant planned airport, instead it has to find the closest airport to its current position that it can make it with its remaining fuel\n", + "A plane traveling across the country suddenly discovers it is low on fuel! It can no longer make it to the distant planned airport, instead it has to find the closest airport to its current position that it can make it with its remaining fuel.\n", "\n", "- Determine the distance of a point to a great circle arc (cross-track and along-track distance)\n", "- Determine if a point lies on a great circle arc and path (with and without tolerances)\n", @@ -281,7 +281,7 @@ "- Cross track distance: angular distance from point P to great circle path\n", "- Along track distance: angular distance along the great circle path from A to B before hitting a point that is closest to point P\n", "\n", - "Cross-Track Distance, sometimes known as cross track error, can also be determined with vectors (typically simpler too)" + "Cross-Track Distance, sometimes known as cross track error, can also be determined with vectors (typically simpler too)." ] }, { @@ -696,7 +696,7 @@ "source": [ "## Determine if a point lies on a great circle arc and path\n", "\n", - "With and without tolerances (in meters)" + "With and without tolerances (in meters):" ] }, { @@ -879,11 +879,11 @@ "metadata": {}, "source": [ "## Summary\n", - "Calculating and plotting the cross-track and along-trackd distance of a great circle arc/path and a point\n", + "Calculating and plotting the cross-track and along-trackd distance of a great circle arc/path and a point.\n", "\n", "### What's next?\n", "\n", - "Determine when a great circle path crosses a given parallel and the maximum and minimum latitude coordinates of a great circle path" + "Determine when a great circle path crosses a given parallel and the maximum and minimum latitude coordinates of a great circle path." ] }, { @@ -918,7 +918,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.8" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/_sources/notebooks/tutorials/3_parallels_max_min.ipynb b/_preview/1/_sources/notebooks/tutorials/3_parallels_max_min.ipynb index ffe1b3b..1015bf7 100644 --- a/_preview/1/_sources/notebooks/tutorials/3_parallels_max_min.ipynb +++ b/_preview/1/_sources/notebooks/tutorials/3_parallels_max_min.ipynb @@ -27,7 +27,7 @@ "source": [ "## Overview\n", "\n", - "A valid great circle path (that is not a path around the equator) will cross a maximum and minimum latitude\n", + "A valid great circle path (that is not a path around the equator) will cross a maximum and minimum latitude.\n", "\n", "- Determine the maximum latitude on a Great Circle Path\n", "- Determine the minimum latitude on a Great Great path\n", @@ -250,9 +250,9 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We have previous determined an equation to derive a great circle path from intermediate points from two points on a great circle arc\n", + "We have previous determined an equation to derive a great circle path from intermediate points from two points on a great circle arc.\n", "\n", - "Without additional calculations, finding the maximum location of the maximum and minimum from the list of points along the great circle path. By default, the function uses 360 points along longitude, so the output will only have a resolution of 1 degree. However, by defining the longitude with more points, the resolution increases" + "Without additional calculations, finding the maximum location of the maximum and minimum from the list of points along the great circle path. By default, the function uses 360 points along longitude, so the output will only have a resolution of 1 degree. However, by defining the longitude with more points, the resolution increases." ] }, { @@ -463,11 +463,11 @@ "\n", "$$\\text{max latitude} = acos(|sin(θ) * cos(φ)|)$$\n", "\n", - "For the purpose of this example, we will use `pyproj` geodesic to determine the bearing based on a great circle arc, but consult previous chapters if you want to determine bearing mathetically based on the unit sphere instead of the ellipsoid\n", + "For the purpose of this example, we will use `pyproj` geodesic to determine the bearing based on a great circle arc, but consult previous chapters if you want to determine bearing mathetically based on the unit sphere instead of the ellipsoid.\n", "\n", "**Important Note**\n", "\n", - "Clairaut's Formula works from unit sphere, and as a result, is subject to errors (about 3%, about +/- 11 degrees)\n", + "Clairaut's Formula works from unit sphere, and as a result, is subject to errors (about 3%, about +/- 11 degrees).\n", "\n", "- [Ed Williams: Clairaut's Formula](https://edwilliams.org/avform147.htm#Clairaut)" ] @@ -541,7 +541,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Like finding maximum, from a list of great circle path, the smallest latitude can be found by analysing the list for the smallest latitude point" + "Like finding maximum, from a list of great circle path, the smallest latitude can be found by analysing the list for the smallest latitude point." ] }, { @@ -663,13 +663,13 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### Maximumn Latitude from Clairaut's Formula (TODO)\n", + "### Maximumn Latitude from Clairaut's Formula\n", "\n", "To solve for the minimum, the true course should be when 0 and 180 degrees on the unit sphere where for *any* bearing/latitude along the great circle:\n", "\n", "$$\\text{min latitude} = asin(|sin(θ) * cos(φ)|)$$\n", "\n", - "The southernmost point is the antipode to the northernmost (max) latitude" + "The southernmost point is the antipode to the northernmost (max) latitude." ] }, { @@ -714,23 +714,9 @@ "source": [ "## Determine when great circle path cross parallels (TODO)\n", "\n", - "Determine the longitude when a great circle crosses a given latitude parrellel" + "Determine the longitude when a great circle crosses a given latitude parrellel." ] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - }, { "cell_type": "markdown", "metadata": {}, @@ -743,11 +729,11 @@ "metadata": {}, "source": [ "## Summary\n", - "Determine the coordinates when a great circle path crosses a specific parallel as well as the maximumn and minimum latitude coordinates\n", + "Determine the coordinates when a great circle path crosses a specific parallel as well as the maximumn and minimum latitude coordinates.\n", "\n", "### What's next?\n", "\n", - "Intersections of Great Circles" + "Intersections of Great Circles." ] }, { @@ -782,7 +768,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.8" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/_sources/notebooks/tutorials/4_path_intersection.ipynb b/_preview/1/_sources/notebooks/tutorials/4_path_intersection.ipynb index fd8bb8f..66a436f 100644 --- a/_preview/1/_sources/notebooks/tutorials/4_path_intersection.ipynb +++ b/_preview/1/_sources/notebooks/tutorials/4_path_intersection.ipynb @@ -26,7 +26,7 @@ "metadata": {}, "source": [ "## Overview\n", - "A great circle path crosses the entire planet and any two valid great circle paths will always intersect\n", + "A great circle path crosses the entire planet and any two valid great circle paths will always intersect.\n", "\n", "- Find the intersection of two great circle paths (always exists)\n", "- Find the intersection of two great circle arcs (if it exists) (TODO)" @@ -244,7 +244,7 @@ "source": [ "## Find the intersection of two great circle paths\n", "\n", - "The intersection of two great circle paths always exists at two positions on the globe if both paths are valid great circle paths (not meridians)" + "The intersection of two great circle paths always exists at two positions on the globe if both paths are valid great circle paths (not meridians)." ] }, { @@ -532,16 +532,9 @@ "source": [ "## Find the intersection of two great circle arcs (TODO)\n", "\n", - "The intersection of two great circle paths always exists at two positions on the globem but intersections do not always exists along the great circle arcs" + "The intersection of two great circle paths always exists at two positions on the globem but intersections do not always exists along the great circle arcs." ] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - }, { "cell_type": "markdown", "metadata": {}, @@ -591,7 +584,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.8" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/_sources/notebooks/tutorials/5_angles.ipynb b/_preview/1/_sources/notebooks/tutorials/5_angles.ipynb index e54b8d5..8992161 100644 --- a/_preview/1/_sources/notebooks/tutorials/5_angles.ipynb +++ b/_preview/1/_sources/notebooks/tutorials/5_angles.ipynb @@ -26,7 +26,7 @@ "metadata": {}, "source": [ "## Overview\n", - "Angles are formed by the intersection of great circle paths\n", + "Angles are formed by the intersection of great circle paths.\n", "\n", "- Calculate the acute and obtuse angle of two Great Circle paths\n", "- Calculate the Directed Angle of two Great Circle paths based on an intersection point\n", @@ -265,7 +265,7 @@ "source": [ "## Calculate the acute and obtuse angle of two great circle paths\n", "\n", - "The acute and obtuse angle formed by two great circle paths and an intersection point" + "The acute and obtuse angle formed by two great circle paths and an intersection point." ] }, { @@ -335,7 +335,7 @@ "source": [ "## Calculate the Directed Angle of two Great Circle paths based on an intersection point\n", "\n", - "Calculate the directed angle of two great circle paths based on an intersection point\n", + "Calculate the directed angle of two great circle paths based on an intersection point.\n", "\n", "### Overview of Directed Angles\n", "\n", @@ -897,7 +897,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.8" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/_sources/notebooks/tutorials/6_polygon_area.ipynb b/_preview/1/_sources/notebooks/tutorials/6_polygon_area.ipynb index 783eee4..59c83ba 100644 --- a/_preview/1/_sources/notebooks/tutorials/6_polygon_area.ipynb +++ b/_preview/1/_sources/notebooks/tutorials/6_polygon_area.ipynb @@ -26,7 +26,7 @@ "metadata": {}, "source": [ "## Overview\n", - "Determine the calculations of a spherical polygons based on a unit sphere\n", + "Determine the calculations of a spherical polygons based on a unit sphere.\n", "\n", "- Determine clockwise/counterclockwise ordering of points on spherical polygon\n", "- Area and Permieter of quadrilateral patch on a unit sphere\n", @@ -606,7 +606,7 @@ "metadata": {}, "source": [ "### TODO\n", - "Fix invalid overlapping polygon by re-ordering points into a clockwise order" + "Fix invalid overlapping polygon by re-ordering points into a clockwise order." ] }, { @@ -939,13 +939,6 @@ " -130, -60, 20, 60)" ] }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] - }, { "cell_type": "markdown", "metadata": {}, @@ -986,7 +979,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.11.8" + "version": "3.13.1" }, "nbdime-conflicts": { "local_diff": [ diff --git a/_preview/1/notebooks/foundations/1_terminology.html b/_preview/1/notebooks/foundations/1_terminology.html index 6329d6b..3e9b1b4 100644 --- a/_preview/1/notebooks/foundations/1_terminology.html +++ b/_preview/1/notebooks/foundations/1_terminology.html @@ -420,7 +420,7 @@

-

https://www.jpl.nasa.gov/edu/images/news/karin_animation.gif

+

https://d2pn8kiwq2w21t.cloudfront.net/images/karin_animation.width-1024.gif

Great Circle Terminology

@@ -428,7 +428,7 @@

Great Circle Terminology

Great Circles are powerful tools used in the navigation like ships and planes as well as in geoscience for working with and understanding remote sensing via satellites. Great Circle mathematics make use of spherical geometry, where, rather than lines, shapes on a sphere are formed by the intersection of arcs along the curved surface.

While spherical geometry played an important role historically in the fields of astronomy and navigation, its teaching has largely fallen out of favor since the 1950’s making finding comphrenshive resources difficult.

-

This notebook will cover some of the important and unique terminology used when working with Great Circles and spherical geometry

+

This notebook will cover some of the important and unique terminology used when working with Great Circles and spherical geometry.

  1. Spherical Geometry

  2. Great Circles

    3. @@ -455,14 +455,6 @@

    Spherical Trigonometry

    Great Circle USGS

-
-

Spherical Triangles

-

TODO: via Spherical trigonometry (Fajardo)

-
-
-

Napier’s Rules

-

TODO: via Spherical trigonometry (Fajardo)

-

Law of Cosines

@@ -478,7 +470,7 @@

Law of Cosines

Law of Sines

-

The spherical law of sines states that angles A, B, and C be the angles opposite of the sides a, b, c, where

+

The spherical law of sines states that angles A, B, and C be the angles opposite of the sides a, b, c, where:

\[\frac{sin(A)}{sin(a)} = \frac{sin(B)}{sin(b)} = \frac{sin(C)}{sin(c)}\]

Proof of Spherical Law of Sines

@@ -488,7 +480,7 @@

Law of Sines

Great Circle Path vs. Great Circle Arc

-

A great circle is the largest circle that can be formed on the surface of a sphere created by the intersection of a plane that also passes through the center of the sphere

+

A great circle is the largest circle that can be formed on the surface of a sphere created by the intersection of a plane that also passes through the center of the sphere.

All Great Circles:

  • Intersect the center of the Earth

    • @@ -505,10 +497,7 @@

    Great Circle Path vs. Great Circle Arc

    Ellipsoids vs. Spheres

    -

    The earth is not round, instead it is an irregular ellipsoid known as a an oblate spheroid where the poles are slightly flatter. Spherical trigonomeyry assumes a unit sphere, so working on Earth, without additional corrections, spherical measurements will contain up to 0.3% (22 km) when assuming the Earth is a perfect sphere

    -
    -

    TODO: https://gis.stackexchange.com/questions/25494/how-accurate-is-approximating-earth-as-sphere

    -
    +

    The earth is not round, instead it is an irregular ellipsoid known as a an oblate spheroid where the poles are slightly flatter. Spherical trigonomeyry assumes a unit sphere, so working on Earth, without additional corrections, spherical measurements will contain up to 0.3% (22 km) when assuming the Earth is a perfect sphere (see more).

    To account for the error when assuming the Earth is a sphere, there are various geodetic systems and ellipsoids to include in calculations.

@@ -518,7 +507,7 @@

Geodesic and Pythonpyproj: Python interface to PROJ (catographic projections and coordinate transformations library)

  • geopy: Python client for several popular geocoding web services

    • -

    pyproj and geopy both take advantage of different types of (optional) ellipsoids

    +

    pyproj and geopy both take advantage of different types of (optional) ellipsoids.

    import pyproj
@@ -597,9 +586,10 @@ 
    Geodesic and Pythongeopy by default makes use of WGS-84 where “the mean earth radius as defined by the International Union of Geodesy and Geophysics, (2a + b)/3 = 6371.0087714150598 kilometers approx 6371.009 km (for WGS-84), resulting in an error of up to about 0.5%” (geopy)
    
-
    WGS-84 is a unified global ellipsoid model that is used for GPS collected from GPS satellites to calculate extremely preise measurements of the Earth. For the purpose of this notebook, this is the ellipsoid model we will be working with
    
+
    
+
    The standard reference ellipsoid for working with Earth is WGS-84
    
+
    geopy by default makes use of WGS-84 where “the mean earth radius as defined by the International Union of Geodesy and Geophysics, (2a + b)/3 = 6371.0087714150598 kilometers approx 6371.009 km (for WGS-84), resulting in an error of up to about 0.5%” (geopy).
    
+
    WGS-84 is a unified global ellipsoid model that is used for GPS collected from GPS satellites to calculate extremely preise measurements of the Earth. For the purpose of this notebook, this is the ellipsoid model we will be working with.
    
 
    
 
    
 
    print(pyproj.list.get_ellps_map()["WGS84"])
@@ -614,12 +604,13 @@ 
    Geodesic and Python6378137.0 meters, an inverse flattening feature of 298.257223563, and a flattening factor of 0.0033528106647474805
    
+
    WGS-84 is an ellipsoid with a semi-major axis of 6378137.0 meters, an inverse flattening feature of 298.257223563, and a flattening factor of 0.0033528106647474805.
    
 
    Learn more!
    
 
    
+

    Geodesy

    -

    Geodesy is the complex science of measuring the Earth’s “geometric shape, orientation in space, and gravity field”

    +

    Geodesy is the complex science of measuring the Earth’s “geometric shape, orientation in space, and gravity field”.

    Learn more!

    Geodesic

    @@ -648,8 +639,8 @@

    An Important Note on Resources

    Summary

    -

    A great circle is formed by a plane intersecting a sphere and the center, like the equator

    -

    Great Circles make use of spherical geometry to measure features on the curved surface of a unit sphere. However, planets like Earth are not perfect spheres and to account for the error are combined with geodesic calculations to reduce the error in final calculations

    +

    A great circle is formed by a plane intersecting a sphere and the center, like the equator.

    +

    Great Circles make use of spherical geometry to measure features on the curved surface of a unit sphere. However, planets like Earth are not perfect spheres and to account for the error are combined with geodesic calculations to reduce the error in final calculations.

    What’s next?

    Coordinates and Great Circles

    @@ -726,8 +717,6 @@

    What’s next?Prerequisites
  • Spherical Geometry @@ -737,7 +726,10 @@

    What’s next?Ellipsoids vs. Spheres

    • -
  • Geodesic and Python
    • +
  • Geodesic and Python +
  • Geodesy diff --git a/_preview/1/notebooks/foundations/2_coordinates.html b/_preview/1/notebooks/foundations/2_coordinates.html index 738978b..9ba21bf 100644 --- a/_preview/1/notebooks/foundations/2_coordinates.html +++ b/_preview/1/notebooks/foundations/2_coordinates.html @@ -426,7 +426,7 @@

    Coordinate Types

    Overview

    -

    Understanding different types of coordinates for working with unit spheres and ellipsoids

    +

    Understanding different types of coordinates for working with unit spheres and ellipsoids.

    1. Types of Coordinates

    2. Convert Coordinates to All Coordinate Types

      3. @@ -484,22 +484,19 @@

      ImportsTypes of Coordinates

      Geodesic Coordinates

      -
      -

      A geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude (Wikipedia)

      -
      -

      Geodesic coordinates are latitude and longtiude, from -90° South to 90° North and -180° East to 180° West measured from Greenwich

      +

      Geodesic coordinates are latitude and longtiude, from -90° South to 90° North and -180° East to 180° West measured from Greenwich.

      Longitude lines are perpendicular to and latitude lines are parallel to the Equator from Wikipedia

      Cartesian Coordinates

      -

      Cartesian coordinates describe points in space based on perpendicular axis lines that meet at a singlle point of origin, where any point’s position is described based on the distance to the origin along xyz axis

      +

      Cartesian coordinates describe points in space based on perpendicular axis lines that meet at a single point of origin, where any point’s position is described based on the distance to the origin along xyz axis.

      A three dimensional Cartesian coordinate system, with origin O and axis lines X, Y and Z, oriented as shown by the arrows. The tick marks on the axes are one length unit apart. The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2, 3, 4) from Wikipedia

      Image Source: Three Dimensional Cartesian Coordinate System

      Geodesic to Cartesian Coordinates

      -

      Assuming the earth’s radius is 6378137 meters

      +

      Assuming the earth’s radius is 6378137 meters:

      \[x = radius * cos(latitude) * cos(longitude)\]
      @@ -523,7 +520,7 @@

      Cartesian Coordinates

      Spherical Coordinates

      -

      Spherical coordinates describe points in space based on three values: radial distance (rho, r) along the radial line between point and the origin, polar angle (theta, θ) between the radial line and the polar axis, and azimuth angle (phi, φ) which is the angle of rotation of the radial line around the polar axis. With a fixed radius, the 3-point coordinates (r, θ, φ) provide a coordinate along a sphere

      +

      Spherical coordinates describe points in space based on three values: radial distance (rho, r) along the radial line between point and the origin, polar angle (theta, θ) between the radial line and the polar axis, and azimuth angle (phi, φ) which is the angle of rotation of the radial line around the polar axis. With a fixed radius, the 3-point coordinates (r, θ, φ) provide a coordinate along a sphere.

      • Radial distance: distance from center to surface of sphere

      • Polar angle: angle between radial line and polar axis

        • @@ -561,8 +558,8 @@

        Spherical Coordinates

        Polar Coordinates

        -

        Polar coordinates are a combination of latitude, longitude, and altitude from the center of the sphere (based on the radius)

        -

        Assuming the earth’s radius is 6378137 meters

        +

        Polar coordinates are a combination of latitude, longitude, and altitude from the center of the sphere (based on the radius).

        +

        Assuming the earth’s radius is 6378137 meters:

        \[x = cos(latitude) * cos(longitude) * radius\]
        @@ -589,7 +586,7 @@

        Polar Coordinates

        Display Coordinates of Cities

        -

        Read in latitude and longitude coordinates from locations

        +

        First we will read in the latitude and longitude coordinates from locations:

        import pandas as pd
@@ -1243,8 +1240,8 @@ 
        United States Map
 
        
 
        Summary
        
-
        Coordinates on the Earth are measured in many different types of coordinate systems: Geodesic (latitude/longitude), cartesian, spherical, and polar. These coordinates will make future calculations simpler by converting a 2D coordinate like latitude/longitude into a 3D space that can be used for vector calculations
        
-
        In Python, coordinates can be mapped on to a world map via Matploblib and Cartopy
        
+
        Coordinates on the Earth are measured in many different types of coordinate systems: Geodesic (latitude/longitude), cartesian, spherical, and polar. These coordinates will make future calculations simpler by converting a 2D coordinate like latitude/longitude into a 3D space that can be used for vector calculations.
        
+
        In Python, coordinates can be mapped on to a world map via matplotlib and cartopy.
        
 
        
 
        What’s next?
        
 
        Great Circle arcs and paths
        
diff --git a/_preview/1/notebooks/notebook-template.html b/_preview/1/notebooks/notebook-template.html
index a7950b8..b99561f 100644
--- a/_preview/1/notebooks/notebook-template.html
+++ b/_preview/1/notebooks/notebook-template.html
@@ -537,8 +537,8 @@ 
        of further and further
 
        header levels
        
 
        as well \(m = a * t / h\) text! Similarly, you have access to other \(\LaTeX\) equation functionality via MathJax (demo below from link),
        
-
        
-()\[\begin{align}
+
        
+()\[\begin{align}
 \dot{x} & = \sigma(y-x) \\
 \dot{y} & = \rho x - y - xz \\
 \dot{z} & = -\beta z + xy
diff --git a/_preview/1/notebooks/tutorials/1_arc_path.html b/_preview/1/notebooks/tutorials/1_arc_path.html
index b9014a9..f443be6 100644
--- a/_preview/1/notebooks/tutorials/1_arc_path.html
+++ b/_preview/1/notebooks/tutorials/1_arc_path.html
@@ -426,8 +426,8 @@ 
        Great Circle Arcs and Path
 
        
 
        Overview
        
-
        Imagine a plane, flying from Cario to Hong Kong. To a passenger, the plane appears to travel a straight path, but the plane actually curves around the surface, held down by the gravity of the planet
        
-
        Great circles are circles that circumnavigate the globe
        
+
        Imagine a plane flying from Cario to Hong Kong. To a passenger, the plane appears to travel a straight path, but the plane actually curves around the surface, held down by the gravity of the planet.
        
+
        Great circles are circles that circumnavigate the globe.
        
 
          
 
        • Distance between Points on a Great Circle Arc
          • 
 
        • Spherical Distance to Degrees
          • 
@@ -624,10 +624,10 @@ 
          ImportsDistance Between Points on a Great Circle Arc
          
 
          
 
          Determine Distance Mathematically via Unit Sphere
          
-
          Distance between point A (latA, lonA) and point B (latB, lonB)
          
+
          Distance between point A (latA, lonA) and point B (latB, lonB):
          
 
          
 \[d=acos(sin(latA)*sin(latB)+cos(latA)*cos(latB)*cos(lonA-lonB))\]
          
-
          For shorter distances (with less rounding errors)
          
+
          For shorter distances (with less rounding errors):
          
 
          
 \[d=2*asin(\sqrt{sin(\frac{latA-latB}{2})^2 + cos(latA)*cos(latB)*sin(\frac{lonA-lonB}{2})^2})\]
          
 
            
@@ -663,17 +663,17 @@ 
            Determine Distance Mathematically via Unit Sphere
 
        
 
        
-
        Additional distance measuerments
        
+
        Additional types of distance measuerments:
        
 
          
-
        • Haversine (TODO)
          • 
-
        • Vincenty Sphere Great Circle Distance (TODO)
          • 
-
        • Vincenty Ellipsoid Great Circle Distance (TODO)
          • 
-
        • Meeus Great Circle Distance (TODO)
          • 
+
        • Haversine
          • 
+
        • Vincenty Sphere Great Circle Distance
          • 
+
        • Vincenty Ellipsoid Great Circle Distance
          • 
+
        • Meeus Great Circle Distance
          • 
 
        
 
        
 
        
 
        Determine Distance Points via Python Package pyproj
        
-
        pyproj accounts for different ellipsoids like WGS-84
        
+
        pyproj accounts for different ellipsoids like WGS-84.
        
 
        First, setup a ellipsoid to represent the Earth (“WGS-84”):
        
 
        
 
        
@@ -700,12 +700,12 @@ 
        Determine Distance Points via Python Package DIA to Logan) which has a distance of 2823 km, using Denver instead of Boulder
        
+
        Compared to the distance from the associated airports in Denver and Boston (DIA to Logan) which has a distance of 2823 km, using Denver instead of Boulder.
        
 
        
 
        
 
        
 
        Spherical Distance to Degrees
        
-
        Convert a distance from meters to degrees, measured along the great circle sphere with a constant spherical radius of ~6371 km (mean radius of Earth)
        
+
        Convert a distance from meters to degrees, measured along the great circle sphere with a constant spherical radius of ~6371 km (mean radius of Earth).
        
 
@@ -742,7 +742,7 @@ 
        Spherical Distance to Degrees
        
 
        
 
        Determine the Bearing Mathematically via Unit Sphere
        
-
        Bearing between point A (latA, lonA) and point B (latB, lonB)
        
+
        Bearing between point A (latA, lonA) and point B (latB, lonB):
        
 
        
 \[x = cos(latA) * sin(latB) - sin(latA) * cos(latB) * cos(lonB - lonA)\]
        
 
        
@@ -771,7 +771,7 @@ 
        Determine the Bearing Mathematically via Unit Sphere
 
        
 
        Determine the Bearing via Python Package pyproj
        
-
        pyproj accounts for different ellipsoids like WGS-84
        
+
        pyproj accounts for different ellipsoids like WGS-84:
        
 
        
 
        
 
        def bearing_between_points_ellps(start_point=None, end_point=None):
@@ -808,7 +808,7 @@ 
        Determine the Bearing via Python Package 
        
 
        
 
        Determine Intermediate Points Mathemetically via Unit Sphere
        
-
        Determine the points (lat, lon) a given fraction of a distance (d) between a starting points A (latA, lonA) and the final point B (latB, lonB) where f is a fraction along the great circle arc. f=0 is point A and f=1 is point B
        
+
        Determine the points (lat, lon) a given fraction of a distance (d) between a starting points A (latA, lonA) and the final point B (latB, lonB) where f is a fraction along the great circle arc. f=0 is point A and f=1 is point B.
        
 
        
 
        Note: The points cannot be antipodal because the path is undefined
        
 
        
@@ -1223,7 +1223,7 @@ 
        Interpolate a fractional distance along arc
 
        Determine the Midpoint of a Great Circle Arc
        
-
        The midpoint of an arc can be determiend as a fractional distance along an arc where f = 0.5
        
+
        The midpoint of an arc can be determiend as a fractional distance along an arc where f = 0.5.
        
 
        
 
        
 
        midpoint = distance_meter / 2
@@ -1286,7 +1286,7 @@ 
        Determine the Midpoint of a Great Circle Arc
 
        Generate a Great Circle Path
        
-
        Get points on the Great Cricle defined by two points
        
+
        Get points on the Great Cricle defined by two points.
        
 
        
 
        
 
        # Generate Latitude Coordinates based on Longitude Coordinates
@@ -1444,7 +1444,7 @@ 
        Generate a Great Circle Path
 
        Determine an Antipodal Point
        
-
        Antipodal is the point on the globe that is on the exact opposite side of the Earth
        
+
        Antipodal is the point on the globe that is on the exact opposite side of the Earth.
        
 
        Antipodal latitude is defined as:
        
 
        
 \[\text{antipodal latitude} = -1 * \text{latitude}\]
        
@@ -1560,10 +1560,10 @@ 
        Determine an Antipodal Point
 
        
 
        Summary
        
-
        Calculating and mapping the midpoint and intermediate points along the great circle arc and closed circle path
        
+
        Calculating and mapping the midpoint and intermediate points along the great circle arc and closed circle path.
        
 
        
 
        What’s next?
        
-
        With a great circle arc defined, determine if a third point is along the arc or at what distance it sits from the great circle arc and path
        
+
        With a great circle arc defined, determine if a third point is along the arc or at what distance it sits from the great circle arc and path.
        
 
        
 
        
 
        
diff --git a/_preview/1/notebooks/tutorials/2_arc_to_point.html b/_preview/1/notebooks/tutorials/2_arc_to_point.html
index e88add7..9ee3202 100644
--- a/_preview/1/notebooks/tutorials/2_arc_to_point.html
+++ b/_preview/1/notebooks/tutorials/2_arc_to_point.html
@@ -426,7 +426,7 @@ 
        Great Circles and a Point
 
        
 
        Overview
        
-
        A plane traveling across the country suddenly discovers it is low on fuel! It can no longer make it to the distant planned airport, instead it has to find the closest airport to its current position that it can make it with its remaining fuel
        
+
        A plane traveling across the country suddenly discovers it is low on fuel! It can no longer make it to the distant planned airport, instead it has to find the closest airport to its current position that it can make it with its remaining fuel.
        
 
-
        Cross-Track Distance, sometimes known as cross track error, can also be determined with vectors (typically simpler too)
        
+
        Cross-Track Distance, sometimes known as cross track error, can also be determined with vectors (typically simpler too).
        
 
        
 
        Cross Track Distance
        
 
        Distance of a point to a great circle arc
        
@@ -955,7 +955,7 @@ 
        Negative Cross-Track Distance: Point lies in the hemiphere to the right of t
 
        
 
        
 
        Determine if a point lies on a great circle arc and path
        
-
        With and without tolerances (in meters)
        
+
        With and without tolerances (in meters):
        
 
        
 
        
 
        def is_point_on_arc(start_point=None, end_point=None,
@@ -1089,10 +1089,10 @@ 
        Check if a point lies on a great circle arc
 
        
 
        Summary
        
-
        Calculating and plotting the cross-track and along-trackd distance of a great circle arc/path and a point
        
+
        Calculating and plotting the cross-track and along-trackd distance of a great circle arc/path and a point.
        
 
        
 
        What’s next?
        
-
        Determine when a great circle path crosses a given parallel and the maximum and minimum latitude coordinates of a great circle path
        
+
        Determine when a great circle path crosses a given parallel and the maximum and minimum latitude coordinates of a great circle path.
        
 
        
 
        
 
        
diff --git a/_preview/1/notebooks/tutorials/3_parallels_max_min.html b/_preview/1/notebooks/tutorials/3_parallels_max_min.html
index 2653247..4fa7c94 100644
--- a/_preview/1/notebooks/tutorials/3_parallels_max_min.html
+++ b/_preview/1/notebooks/tutorials/3_parallels_max_min.html
@@ -426,7 +426,7 @@ 
        Great Circles and Parallels
 
        
 
        Overview
        
-
        A valid great circle path (that is not a path around the equator) will cross a maximum and minimum latitude
        
+
        A valid great circle path (that is not a path around the equator) will cross a maximum and minimum latitude.
        
 
          
 
        • Determine the maximum latitude on a Great Circle Path
          • 
 
        • Determine the minimum latitude on a Great Great path
          • 
@@ -618,8 +618,8 @@ 
          Imports
 
          
 
          Maximum Latitude on a Great Circle Path
          
-
          We have previous determined an equation to derive a great circle path from intermediate points from two points on a great circle arc
          
-
          Without additional calculations, finding the maximum location of the maximum and minimum from the list of points along the great circle path. By default, the function uses 360 points along longitude, so the output will only have a resolution of 1 degree. However, by defining the longitude with more points, the resolution increases
          
+
          We have previous determined an equation to derive a great circle path from intermediate points from two points on a great circle arc.
          
+
          Without additional calculations, finding the maximum location of the maximum and minimum from the list of points along the great circle path. By default, the function uses 360 points along longitude, so the output will only have a resolution of 1 degree. However, by defining the longitude with more points, the resolution increases.
          
 
          
 
          
 
          # Generate Latitude Coordinates based on Longitude Coordinates
@@ -778,9 +778,9 @@ 
          Maximumn Latitude from Clairaut’s Formula
 \[\text{max latitude} = acos(|sin(θ) * cos(φ)|)\]
          
-
          For the purpose of this example, we will use pyproj geodesic to determine the bearing based on a great circle arc, but consult previous chapters if you want to determine bearing mathetically based on the unit sphere instead of the ellipsoid
          
+
          For the purpose of this example, we will use pyproj geodesic to determine the bearing based on a great circle arc, but consult previous chapters if you want to determine bearing mathetically based on the unit sphere instead of the ellipsoid.
          
 
          Important Note
          
-
          Clairaut’s Formula works from unit sphere, and as a result, is subject to errors (about 3%, about +/- 11 degrees)
          
+
          Clairaut’s Formula works from unit sphere, and as a result, is subject to errors (about 3%, about +/- 11 degrees).
          
           
@@ -832,7 +832,7 @@ 
          Antipodal Point of Max (TODO)
 
          
 
          gc_lat_lon = generate_latitude_along_gc("boulder", "houston", number_of_lon_pts=360)
@@ -908,12 +908,12 @@ 
          Minimum Latitude along Great Circle Path
-
          Maximumn Latitude from Clairaut’s Formula (TODO)
          
+
          
+
          Maximumn Latitude from Clairaut’s Formula
          
 
          To solve for the minimum, the true course should be when 0 and 180 degrees on the unit sphere where for any bearing/latitude along the great circle:
          
 
          
 \[\text{min latitude} = asin(|sin(θ) * cos(φ)|)\]
          
-
          The southernmost point is the antipode to the northernmost (max) latitude
          
+
          The southernmost point is the antipode to the northernmost (max) latitude.
          
 
          
 
          
 
          def clairaut_formula_min(start_point=None, end_point=None):
@@ -947,15 +947,15 @@ 
          Maximumn Latitude from Clairaut’s Formula (TODO)
 
          
 
          Determine when great circle path cross parallels (TODO)
          
-
          Determine the longitude when a great circle crosses a given latitude parrellel
          
+
          Determine the longitude when a great circle crosses a given latitude parrellel.
          
 
          
 
          
 
          
 
          Summary
          
-
          Determine the coordinates when a great circle path crosses a specific parallel as well as the maximumn and minimum latitude coordinates
          
+
          Determine the coordinates when a great circle path crosses a specific parallel as well as the maximumn and minimum latitude coordinates.
          
 
          
 
          What’s next?
          
-
          Intersections of Great Circles
          
+
          Intersections of Great Circles.
          
 
          
 
          
 
          
@@ -1043,7 +1043,7 @@ 
          Resources and referencesMinimum Latitude on a Great Circle Path
 
 
        • Determine when great circle path cross parallels (TODO)
          • 
diff --git a/_preview/1/notebooks/tutorials/4_path_intersection.html b/_preview/1/notebooks/tutorials/4_path_intersection.html
index 5b58232..4265ad2 100644
--- a/_preview/1/notebooks/tutorials/4_path_intersection.html
+++ b/_preview/1/notebooks/tutorials/4_path_intersection.html
@@ -424,7 +424,7 @@ 
          Intersections of Great Circles
 
          
 
          Overview
          
-
          A great circle path crosses the entire planet and any two valid great circle paths will always intersect
          
+
          A great circle path crosses the entire planet and any two valid great circle paths will always intersect.
          
 
            
 
          • Find the intersection of two great circle paths (always exists)
            • 
 
          • Find the intersection of two great circle arcs (if it exists) (TODO)
            • 
@@ -616,7 +616,7 @@ 
            Imports
 
            
 
            Find the intersection of two great circle paths
            
-
            The intersection of two great circle paths always exists at two positions on the globe if both paths are valid great circle paths (not meridians)
            
+
            The intersection of two great circle paths always exists at two positions on the globe if both paths are valid great circle paths (not meridians).
            
 
            
 
            Math of intersection
            
 
            TODO
            
@@ -854,7 +854,7 @@ 
            Plot Intersections with Great Circle Paths
 
            Find the intersection of two great circle arcs (TODO)
            
-
            The intersection of two great circle paths always exists at two positions on the globem but intersections do not always exists along the great circle arcs
            
+
            The intersection of two great circle paths always exists at two positions on the globem but intersections do not always exists along the great circle arcs.
            
 
            
 
            
 
            
diff --git a/_preview/1/notebooks/tutorials/5_angles.html b/_preview/1/notebooks/tutorials/5_angles.html
index fd47fe2..2cccda6 100644
--- a/_preview/1/notebooks/tutorials/5_angles.html
+++ b/_preview/1/notebooks/tutorials/5_angles.html
@@ -424,7 +424,7 @@ 
            Angles and Great Circles
 
            
 
            Overview
            
-
            Angles are formed by the intersection of great circle paths
            
+
            Angles are formed by the intersection of great circle paths.
            
 
              
 
            • Calculate the acute and obtuse angle of two Great Circle paths
              • 
 
            • Calculate the Directed Angle of two Great Circle paths based on an intersection point
              • 
@@ -629,7 +629,7 @@ 
              Imports
 
              
 
              Calculate the acute and obtuse angle of two great circle paths
              
-
              The acute and obtuse angle formed by two great circle paths and an intersection point
              
+
              The acute and obtuse angle formed by two great circle paths and an intersection point.
              
 
              
 
              
 
              def angle_between_arcs(start_gc1=None, end_gc1=None,
@@ -680,7 +680,7 @@ 
              Calculate the acute and obtuse angle of two great circle paths
 
              Calculate the Directed Angle of two Great Circle paths based on an intersection point
              
-
              Calculate the directed angle of two great circle paths based on an intersection point
              
+
              Calculate the directed angle of two great circle paths based on an intersection point.
              
 
              
 
              Overview of Directed Angles
              
 
              TODO
              
diff --git a/_preview/1/notebooks/tutorials/6_polygon_area.html b/_preview/1/notebooks/tutorials/6_polygon_area.html
index feb1290..895efbc 100644
--- a/_preview/1/notebooks/tutorials/6_polygon_area.html
+++ b/_preview/1/notebooks/tutorials/6_polygon_area.html
@@ -423,7 +423,7 @@ 
              Spherical Polygons and Areas
 
              
 
              Overview
              
-
              Determine the calculations of a spherical polygons based on a unit sphere
              
+
              Determine the calculations of a spherical polygons based on a unit sphere.
              
 
                
 
              • Determine clockwise/counterclockwise ordering of points on spherical polygon
                • 
 
              • Area and Permieter of quadrilateral patch on a unit sphere
                • 
@@ -861,7 +861,7 @@ 
                Area and Perimeter of quadrilateral patch
 
                TODO
                
-
                Fix invalid overlapping polygon by re-ordering points into a clockwise order
                
+
                Fix invalid overlapping polygon by re-ordering points into a clockwise order.
                
 
                
 
                
 
                plot_area(["boulder", "boston", "houston", "boston", "cairo", "arecibo", "greenwich"])
diff --git a/_preview/1/searchindex.js b/_preview/1/searchindex.js
index b2021ef..5ddc2d8 100644
--- a/_preview/1/searchindex.js
+++ b/_preview/1/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"alltitles": {"1. Great Circle Arcs and Paths": [[0, "great-circle-arcs-and-paths"]], "2. Great Circles and a Point": [[0, "great-circles-and-a-point"]], "3. Great Circles and Parallels": [[0, "great-circles-and-parallels"]], "4. Intersections of Great Circles": [[0, "intersections-of-great-circles"]], "5. Angles and Great Circles": [[0, "angles-and-great-circles"]], "6. Spherical Polygons and Areas": [[0, "spherical-polygons-and-areas"]], "A content subsection": [[4, "a-content-subsection"]], "Add Columns for Additional Coordinate Types": [[2, "add-columns-for-additional-coordinate-types"]], "Along Track Distance": [[6, "along-track-distance"]], "An Important Note on Resources": [[1, "an-important-note-on-resources"]], "Angles and Great Circles": [[9, null]], "Another content subsection": [[4, "another-content-subsection"]], "Antipodal Point of Max (TODO)": [[7, "antipodal-point-of-max-todo"]], "Area and Perimeter of quadrilateral patch": [[10, "area-and-perimeter-of-quadrilateral-patch"]], "Authors": [[0, "authors"]], "Calculate Intersection Point Between Two Great Circle Paths": [[9, "calculate-intersection-point-between-two-great-circle-paths"]], "Calculate the Directed Angle of two Great Circle paths based on an intersection point": [[9, "calculate-the-directed-angle-of-two-great-circle-paths-based-on-an-intersection-point"]], "Calculate the acute and obtuse angle of two great circle paths": [[9, "calculate-the-acute-and-obtuse-angle-of-two-great-circle-paths"]], "Cartesian Coordinates": [[2, "cartesian-coordinates"]], "Check if a point lies on a great circle arc": [[6, "check-if-a-point-lies-on-a-great-circle-arc"]], "Contributors": [[0, "contributors"]], "Convert City Coordinates to All Coordinate Types": [[2, "convert-city-coordinates-to-all-coordinate-types"]], "Coordinate Types": [[2, null]], "Cross Track Distance": [[6, "cross-track-distance"]], "Danger": [[4, null]], "Determine Distance Mathematically via Unit Sphere": [[5, "determine-distance-mathematically-via-unit-sphere"]], "Determine Distance Points via Python Package pyproj": [[5, "determine-distance-points-via-python-package-pyproj"]], "Determine Intermediate Points Mathemetically via Unit Sphere": [[5, "determine-intermediate-points-mathemetically-via-unit-sphere"]], "Determine Intermediate Points via Python Package pyproj and geopy": [[5, "determine-intermediate-points-via-python-package-pyproj-and-geopy"]], "Determine an Antipodal Point": [[5, "determine-an-antipodal-point"]], "Determine clockwise/counterclockwise ordering of points on spherical polygon": [[10, "determine-clockwise-counterclockwise-ordering-of-points-on-spherical-polygon"]], "Determine if a given point is within a spherical polygon": [[10, "determine-if-a-given-point-is-within-a-spherical-polygon"]], "Determine if a point lies on a great circle arc and path": [[6, "determine-if-a-point-lies-on-a-great-circle-arc-and-path"]], "Determine the Bearing Mathematically via Unit Sphere": [[5, "determine-the-bearing-mathematically-via-unit-sphere"]], "Determine the Bearing of a Great Circle Arc": [[5, "determine-the-bearing-of-a-great-circle-arc"]], "Determine the Bearing via Python Package pyproj": [[5, "determine-the-bearing-via-python-package-pyproj"]], "Determine the Midpoint of a Great Circle Arc": [[5, "determine-the-midpoint-of-a-great-circle-arc"]], "Determine the distance of a point to a great circle arc": [[6, "determine-the-distance-of-a-point-to-a-great-circle-arc"]], "Determine when great circle path cross parallels (TODO)": [[7, "determine-when-great-circle-path-cross-parallels-todo"]], "Display Coordinates of Cities": [[2, "display-coordinates-of-cities"]], "Distance Between Points on a Great Circle Arc": [[5, "distance-between-points-on-a-great-circle-arc"]], "Ellipsoids vs. Spheres": [[1, "ellipsoids-vs-spheres"]], "Find the intersection of two great circle arcs (TODO)": [[8, "find-the-intersection-of-two-great-circle-arcs-todo"]], "Find the intersection of two great circle paths": [[8, "find-the-intersection-of-two-great-circle-paths"]], "Foundations and Terminology": [[0, "foundations-and-terminology"]], "Generate Points at Intermediate Points along an Arc/Path": [[6, "generate-points-at-intermediate-points-along-an-arc-path"]], "Generate a Great Circle Path": [[5, "generate-a-great-circle-path"]], "Generating a Great Circle Arc with Intermediates Points": [[5, "generating-a-great-circle-arc-with-intermediates-points"]], "Geodesic": [[1, "geodesic"]], "Geodesic Coordinates": [[2, "geodesic-coordinates"]], "Geodesic and Python": [[1, "geodesic-and-python"]], "Geodesy": [[1, "geodesy"]], "Great Circle Arcs and Path": [[5, null]], "Great Circle Cookbook": [[0, null]], "Great Circle Path vs. Great Circle Arc": [[1, "great-circle-path-vs-great-circle-arc"]], "Great Circle Terminology": [[1, null]], "Great Circles": [[1, "great-circles"]], "Great Circles and Parallels": [[7, null]], "Great Circles and a Point": [[6, null]], "How to Cite This Cookbook": [[3, null]], "Imports": [[2, "imports"], [4, "imports"], [5, "imports"], [6, "imports"], [7, "imports"], [8, "imports"], [9, "imports"], [10, "imports"]], "Info": [[4, null]], "Interpolate a fractional distance along arc": [[5, "interpolate-a-fractional-distance-along-arc"]], "Interpolate every N meters": [[5, "interpolate-every-n-meters"]], "Interpolate with N Total Equally Spaced Points": [[5, "interpolate-with-n-total-equally-spaced-points"]], "Intersections of Great Circles": [[8, null]], "Last Section": [[4, "last-section"]], "Law of Cosines": [[1, "law-of-cosines"]], "Law of Sines": [[1, "law-of-sines"]], "Math of intersection": [[8, "math-of-intersection"]], "Maximum Latitude on a Great Circle Path": [[7, "maximum-latitude-on-a-great-circle-path"]], "Maximumn Latitude from Clairaut\u2019s Formula": [[7, "maximumn-latitude-from-clairauts-formula"]], "Maximumn Latitude from Clairaut\u2019s Formula (TODO)": [[7, "maximumn-latitude-from-clairauts-formula-todo"]], "Mean center of spherical polygon": [[10, "mean-center-of-spherical-polygon"]], "Minimum Latitude along Great Circle Path": [[7, "minimum-latitude-along-great-circle-path"]], "Minimum Latitude on a Great Circle Path": [[7, "minimum-latitude-on-a-great-circle-path"]], "Motivation": [[0, "motivation"]], "Napier\u2019s Rules": [[1, "napiers-rules"]], "Negative Cross-Track Distance: Point lies in the hemiphere to the right of the great circle": [[6, "negative-cross-track-distance-point-lies-in-the-hemiphere-to-the-right-of-the-great-circle"]], "Overview": [[1, "overview"], [2, "overview"], [4, "overview"], [5, "overview"], [6, "overview"], [7, "overview"], [8, "overview"], [9, "overview"], [10, "overview"]], "Overview of Directed Angles": [[9, "overview-of-directed-angles"]], "Plot Arcs as Points on a World Map": [[5, "plot-arcs-as-points-on-a-world-map"]], "Plot Coordinates": [[2, "plot-coordinates"]], "Plot Directed Angle": [[9, "plot-directed-angle"]], "Plot Intersections with Great Circle Paths": [[8, "plot-intersections-with-great-circle-paths"]], "Plot Maximum": [[7, "plot-maximum"]], "Polar Coordinates": [[2, "polar-coordinates"]], "Positive Cross-Track Distance: Point lies in the hemisphere to the left of the great circle": [[6, "positive-cross-track-distance-point-lies-in-the-hemisphere-to-the-left-of-the-great-circle"]], "Prerequisites": [[1, 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\ No newline at end of file
+Search.setIndex({"alltitles": {"1. Great Circle Arcs and Paths": [[0, "great-circle-arcs-and-paths"]], "2. Great Circles and a Point": [[0, "great-circles-and-a-point"]], "3. Great Circles and Parallels": [[0, "great-circles-and-parallels"]], "4. Intersections of Great Circles": [[0, "intersections-of-great-circles"]], "5. Angles and Great Circles": [[0, "angles-and-great-circles"]], "6. Spherical Polygons and Areas": [[0, "spherical-polygons-and-areas"]], "A content subsection": [[4, "a-content-subsection"]], "Add Columns for Additional Coordinate Types": [[2, "add-columns-for-additional-coordinate-types"]], "Along Track Distance": [[6, "along-track-distance"]], "An Important Note on Resources": [[1, "an-important-note-on-resources"]], "Angles and Great Circles": [[9, null]], "Another content subsection": [[4, "another-content-subsection"]], "Antipodal Point of Max (TODO)": [[7, "antipodal-point-of-max-todo"]], "Area and Perimeter of quadrilateral patch": [[10, "area-and-perimeter-of-quadrilateral-patch"]], "Authors": [[0, "authors"]], "Calculate Intersection Point Between Two Great Circle Paths": [[9, "calculate-intersection-point-between-two-great-circle-paths"]], "Calculate the Directed Angle of two Great Circle paths based on an intersection point": [[9, "calculate-the-directed-angle-of-two-great-circle-paths-based-on-an-intersection-point"]], "Calculate the acute and obtuse angle of two great circle paths": [[9, "calculate-the-acute-and-obtuse-angle-of-two-great-circle-paths"]], "Cartesian Coordinates": [[2, "cartesian-coordinates"]], "Check if a point lies on a great circle arc": [[6, "check-if-a-point-lies-on-a-great-circle-arc"]], "Contributors": [[0, "contributors"]], "Convert City Coordinates to All Coordinate Types": [[2, "convert-city-coordinates-to-all-coordinate-types"]], "Coordinate Types": [[2, null]], "Cross Track Distance": [[6, "cross-track-distance"]], "Danger": [[4, null]], "Determine Distance Mathematically via Unit Sphere": [[5, "determine-distance-mathematically-via-unit-sphere"]], "Determine Distance Points via Python Package pyproj": [[5, "determine-distance-points-via-python-package-pyproj"]], "Determine Intermediate Points Mathemetically via Unit Sphere": [[5, "determine-intermediate-points-mathemetically-via-unit-sphere"]], "Determine Intermediate Points via Python Package pyproj and geopy": [[5, "determine-intermediate-points-via-python-package-pyproj-and-geopy"]], "Determine an Antipodal Point": [[5, "determine-an-antipodal-point"]], "Determine clockwise/counterclockwise ordering of points on spherical polygon": [[10, "determine-clockwise-counterclockwise-ordering-of-points-on-spherical-polygon"]], "Determine if a given point is within a spherical polygon": [[10, "determine-if-a-given-point-is-within-a-spherical-polygon"]], "Determine if a point lies on a great circle arc and path": [[6, "determine-if-a-point-lies-on-a-great-circle-arc-and-path"]], "Determine the Bearing Mathematically via Unit Sphere": [[5, "determine-the-bearing-mathematically-via-unit-sphere"]], "Determine the Bearing of a Great Circle Arc": [[5, "determine-the-bearing-of-a-great-circle-arc"]], "Determine the Bearing via Python Package pyproj": [[5, "determine-the-bearing-via-python-package-pyproj"]], "Determine the Midpoint of a Great Circle Arc": [[5, "determine-the-midpoint-of-a-great-circle-arc"]], "Determine the distance of a point to a great circle arc": [[6, "determine-the-distance-of-a-point-to-a-great-circle-arc"]], "Determine when great circle path cross parallels (TODO)": [[7, "determine-when-great-circle-path-cross-parallels-todo"]], "Display Coordinates of Cities": [[2, "display-coordinates-of-cities"]], "Distance Between Points on a Great Circle Arc": [[5, "distance-between-points-on-a-great-circle-arc"]], "Ellipsoids vs. Spheres": [[1, "ellipsoids-vs-spheres"]], "Find the intersection of two great circle arcs (TODO)": [[8, "find-the-intersection-of-two-great-circle-arcs-todo"]], "Find the intersection of two great circle paths": [[8, "find-the-intersection-of-two-great-circle-paths"]], "Foundations and Terminology": [[0, "foundations-and-terminology"]], "Generate Points at Intermediate Points along an Arc/Path": [[6, "generate-points-at-intermediate-points-along-an-arc-path"]], "Generate a Great Circle Path": [[5, "generate-a-great-circle-path"]], "Generating a Great Circle Arc with Intermediates Points": [[5, "generating-a-great-circle-arc-with-intermediates-points"]], "Geodesic": [[1, "geodesic"]], "Geodesic Coordinates": [[2, "geodesic-coordinates"]], "Geodesic and Python": [[1, "geodesic-and-python"]], "Geodesy": [[1, "geodesy"]], "Great Circle Arcs and Path": [[5, null]], "Great Circle Cookbook": [[0, null]], "Great Circle Path vs. Great Circle Arc": [[1, "great-circle-path-vs-great-circle-arc"]], "Great Circle Terminology": [[1, null]], "Great Circles": [[1, "great-circles"]], "Great Circles and Parallels": [[7, null]], "Great Circles and a Point": [[6, null]], "How to Cite This Cookbook": [[3, null]], "Imports": [[2, "imports"], [4, "imports"], [5, "imports"], [6, "imports"], [7, "imports"], [8, "imports"], [9, "imports"], [10, "imports"]], "Info": [[4, null]], "Interpolate a fractional distance along arc": [[5, "interpolate-a-fractional-distance-along-arc"]], "Interpolate every N meters": [[5, "interpolate-every-n-meters"]], "Interpolate with N Total Equally Spaced Points": [[5, "interpolate-with-n-total-equally-spaced-points"]], "Intersections of Great Circles": [[8, null]], "Last Section": [[4, "last-section"]], "Law of Cosines": [[1, "law-of-cosines"]], "Law of Sines": [[1, "law-of-sines"]], "Math of intersection": [[8, "math-of-intersection"]], "Maximum Latitude on a Great Circle Path": [[7, "maximum-latitude-on-a-great-circle-path"]], "Maximumn Latitude from Clairaut\u2019s Formula": [[7, "maximumn-latitude-from-clairauts-formula"], [7, "id1"]], "Mean center of spherical polygon": [[10, "mean-center-of-spherical-polygon"]], "Minimum Latitude along Great Circle Path": [[7, "minimum-latitude-along-great-circle-path"]], "Minimum Latitude on a Great Circle Path": [[7, "minimum-latitude-on-a-great-circle-path"]], "Motivation": [[0, "motivation"]], "Negative Cross-Track Distance: Point lies in the hemiphere to the right of the great circle": [[6, "negative-cross-track-distance-point-lies-in-the-hemiphere-to-the-right-of-the-great-circle"]], "Overview": [[1, "overview"], [2, "overview"], [4, "overview"], [5, "overview"], [6, "overview"], [7, "overview"], [8, "overview"], [9, "overview"], [10, "overview"]], "Overview of Directed Angles": [[9, "overview-of-directed-angles"]], "Plot Arcs as Points on a World Map": [[5, "plot-arcs-as-points-on-a-world-map"]], "Plot Coordinates": [[2, "plot-coordinates"]], "Plot Directed Angle": [[9, "plot-directed-angle"]], "Plot Intersections with Great Circle Paths": [[8, "plot-intersections-with-great-circle-paths"]], "Plot Maximum": [[7, "plot-maximum"]], "Polar Coordinates": [[2, "polar-coordinates"]], "Positive Cross-Track Distance: Point lies in the hemisphere to the left of the great circle": [[6, "positive-cross-track-distance-point-lies-in-the-hemisphere-to-the-left-of-the-great-circle"]], "Prerequisites": [[1, "prerequisites"], [2, "prerequisites"], [4, "prerequisites"], [5, 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\ No newline at end of file