# Spherical Geometry Environment

- Author:
- Dr. Jack L. Jackson II

- Topic:
- Geometry

## Spherical Geometry Environment

**Be sure to open the activity in the GeoGebra app to have access to the custom tools.**Open the activity above in the app by clicking on the icon with three vertical dots in the upper right of the window and selecting Open in App. This will give an environment for exploring Spherical Geometry on the unit sphere. Spherical Geometry tools analogous to the standard Euclidean tools provided in GeoGebra are provided. You can use this environment to explore Spherical Geometry just as you can use the standard tools to explore Euclidean Geometry.

__Objects Showing Up When You Open the Sketch in the App__**Spherical Plane**(Unit Sphere)

**Checkboxes**in Algebra View to Hide or Show o

**EquatorPoles**Shows equator and reference points on coordinate axes. Points on z-axis and sphere are north and south poles. o

**Longitudes**Shows longitude rays with longitudes having integer multiples of π/12. o

**Latitudes**Shows latitude circles having latitudes with integer multiples of π/12.

**o All points should be in the plane (i.e. on the unit sphere). o You will have to have the points created before using them in tools. o All distance and angle measures should be in the interval [0,π]. Angles are measured in radians. o Latitudes are in the interval [-π/2, π/2]. Latitude 0 is the equator, which is in the unit circle in the**

__Measurements__*xy*-plane. Latitude π/2 is the North Pole (NP), and latitude -π/2 is the South Pole SP. o Longitudes are in the interval (-π, π]. Longitude 0 goes through the point (1,0,0) i.e. the origin of the coordinate system for the plane (0, 0). Longitude π/2 goes through (0,1,0). Longitude π goes through (-1,0,0). Longitude -π/2 goes through (0,-1,0).

**The custom tools have been placed into menus in the toolbar in the following order, which is similar to the standard GeoGebra Euclidean tools for a plane. However, GeoGebra has randomly been rearranging these, so you might have to search for the tools a bit.**

__Custom Spherical Geometry Tools Menus____o__

**Points****Plot Point**Select Latitude and Longitude numbers to plot a point. o

**Latitude & Longitude of Point**Select a point to see its coordinates displayed (latitude, longitude). o

**S Distance**Select two points to compute the spherical distance from one to the other. o

**Antipodal Point**Select a point to generate its antipodal point. o

**S Midpoint**Select two endpoints to generate the midpoint of spherical segment.

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**Line****S Line**Select two non-antipodal points to generate the spherical line containing them. o

**S Ray**Select an endpoint and another point on the ray to generate the spherical ray. o

**S Line Segment**Select two non-antipodal points to generate the spherical line segments with these endpoints. o

**Longitude Line through Point**Select a point to generate the entire line containing the longitude ray through that point. o

**Longitude Ray by Number**Select a number to generate a longitude ray with that measure. o

**Perpendicular to Point on Line**Select a point on a line and another point on a line to generate a line perpendicular to the given line through the first point. o

**Perpendicular through point off line**Select two points on a line and then a point not on the line to produce a line perpendicular to the given line through the last point. o

**Perpendicular Bisector**Select two endpoints of a line segment to produce the perpendicular bisector line for the line segment.

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**Circle****S Circle by Center and Point**Select a center point and a point on a circle to generate the circle. o

**S Circle by Center and Radius**Select a center point and a radius measurement to generate the circle. o

**Latitude Circle through Point**Select a point to generate the latitude circle (not a line) through the point. o

**Latitude Circle by Number**Select a number to generate a latitude circle with that measure.

__o__

**Angle****Angle PVP**Select a point on one ray, the vertex point, and a point on the other ray to generate the angle. o

**Angle Measure**Select a point on one ray, the vertex point, and a point on the other ray to generate the angle measure and marker. o

**Angle of Specified Measure**Select a point on one ray, the vertex point, and an angle measure to produce an angle of this measure. It will rotate in the positive direction. You can select a negative angle measure to rotate in the opposite direction. o

**Angle Bisector**Select a point on one ray, the vertex point, and a point on the other ray to generate the angle bisector.

__o__

**Polygons****S Triangle Measured**Select three points to generate a triangle with these vertices. Angle markers will also be created. Measures of all sides and angles and the excess are also displayed. The excess is the amount the sum of the measures of the interior angles exceeds π.

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**Transformations****S Rotation**Select a preimage point, a center point, and then an amount of rotation to generate an image point. o

**S Reflection**Select a preimage point and then two points on a mirror line to produce the image point.

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**Euclidean Figures****E Tangent Plane**Select a point to produce the Euclidean tangent plane to the unit sphere through that point. o

**E Tangent Ray**Select an endpoint and a point on a Spherical ray to generate the Euclidean tangent ray from the endpoint tangent to the Euclidean semicircle.