The projection onto a tangent plane is a way to represent points from a curved surface, like a sphere, onto a flat surface that touches the curved surface at a single point.
Understanding Tangent Plane Projection
Imagine a flat plane just touching a sphere at one specific point. This plane is called a tangent plane. A projection onto this plane is a method used to map points from the sphere's surface onto this flat surface.
Based on the provided reference, in TAN (tangent) or gnomic geometry, this projection is specifically achieved by following a straight line path.
The Mechanism of Projection
The core principle described for this type of projection involves extending a line from the center of the sphere.
Here's how it works for a specific point on the sphere:
- Locate the point on the sphere you want to project.
- Draw a straight line starting from the exact center of the sphere.
- Extend this line through the point on the sphere.
- Continue extending this line until it intersects the tangent plane.
The point where this line hits the tangent plane is the projection of the original point from the sphere.
Practical Application: Optical Astronomy
This method of projection, extending a line from the center through the point to the tangent plane, is common in optical astronomy. It is used to map celestial objects observed on the seemingly spherical celestial sphere onto a flat photographic plate or digital sensor (which acts as the tangent plane).
This type of projection preserves angles from the perspective of the center of the sphere, making it useful for certain astronomical measurements and charting.
Key Features
- Center-Based: The projection originates from the center of the sphere.
- Straight Lines: Points are mapped along straight lines passing through the center and the point on the sphere.
- Tangent Plane: The target surface is a plane that touches the sphere at one point.
- Distortion: Like most projections, this method introduces distortion, especially for points far away from the point of tangency. Angles from the center are preserved, but distances and areas are significantly distorted further from the center.
In summary, projecting onto a tangent plane using the method described in the reference means representing points from a sphere by drawing lines from the sphere's center through the points and onto a flat plane touching the sphere.
