How does a stereographic projection work?

How does a stereographic projection work?

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. It is conformal, meaning that it preserves angles at which curves meet.

How do you find a stereographic projection?

The stereographic projection of the circle is the set of points Q for which P = s-1(Q) is on the circle, so we substitute the formula for P into the equation for the circle on the sphere to get an equation for the set of points in the projection. P = (1/(1+u2 + v2)[2u, 2v, u2 + v2 – 1] = [x, y, z].

What is a stereographic projection used for?

Stereographic is a planar perspective projection, viewed from the point on the globe opposite the point of tangency. It projects points on a spheroid directly to the plane and it is the only azimuthal conformal projection. The projection is most commonly used in polar aspects for topographic maps of polar regions.

Is stereographic projection a Homeomorphism?

Example: Stereographic Projection Stereographic projection is an important homeomorphism between the plane R 2 \mathbb{R}^2 R2 and the 2 2 2-sphere minus a point.

What is stereographic method?

Stereographic projection is a method used in crystallography and structural geology to depict the angular relationships between crystal faces and geologic structures, respectively. We orient the crystal such that the pole to the (001) face (the c axis) is vertical and points to the North pole of the sphere.

What is great circle in stereographic projection?

The orientation of a plane is represented by imagining the plane to pass through the centre of a sphere (Fig. 1a). The line of intersection between the plane and the sphere will then represent a circle, and this circle is formally known as a great circle.

What is the meaning of stereographic?

: of, relating to, or being a delineation of the form of a solid body (such as the earth) on a plane stereographic projection.

What is stereographic projection in crystallography?

What does stereographic projection preserve?

Stereographic projection preserves circles and angles. That is, the image of a circle on the sphere is a circle in the plane and the angle between two lines on the sphere is the same as the angle between their images in the plane. A projection that preserves angles is called a conformal projection.

What is Gnomonic projection?

Gnomonic is an azimuthal projection that uses the center of the earth as its perspective point. It projects great circles as straight lines, regardless of the aspect. The projection is not conformal nor is it equal-area.

Why is stereographic projection important in crystallography?

The importance of the stereographic projection in crystallography derives from the fact that a set of points on the surface of a sphere provides a complete repre- sentation of a set of directions in three-dimensional space, the directions being the set of lines from the center of the sphere to the set of points.

What is Polar zenithal stereographic projection?

Polar Zenithal Stereographic Projection It is a perspective projection, with the source of light lying at the pole diametrically opposite to one at which the projection plane touches the generating globe.

What is the definition of a stereographic projection?

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles.

Can a stereographic projection miss one point on the sphere?

Although any stereographic projection misses one point on the sphere (the projection point), the entire sphere can be mapped using two projections from distinct projection points. In other words, the sphere can be covered by two stereographic parametrizations (the inverses of the projections) from the plane.

How is stereographic projection used in a Schlegel diagram?

Stereographic projection is also applied to the visualization of polytopes. In a Schlegel diagram, an n-dimensional polytope in R n+1 is projected onto an n-dimensional sphere, which is then stereographically projected onto R n. The reduction from R n+1 to R n can make the polytope easier to visualize and understand.

How are stereographic projections used in Astrolabes?

A second, seemingly unrelated projection is the stereographic projection, not usually to map the earth, but to map the sky. It has been used on astrolabes to measure and display astronomical observations more than 2000 years ago.