# Spherical Geometry Environment

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. Measurements 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 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). Custom Spherical Geometry Tools Menus 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. PointsPlot 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. LineS 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. CircleS 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. AngleAngle 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. PolygonsS 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.   TransformationsS 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. Euclidean FiguresE 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.