Euclidean vs. Spherical vs. Hyperbolic Geometry
Each of These Geometries Has a Different Type of Curvature
Here are the definitions for each type:
- Positive Curvature: A surface has positive curvature at a point if the surface curves away from that point in the same direction relative to the tangent to the surface, regardless of the cutting plane. Alternatively, the surface stays on one side of the tangent plane at that point. (Thus the top of your head, the end of your finger, or the inside of your armpit are points of positive curvature.)
- Negative Curvature: A surface has negative curvature at a point if the surface curves away from the tangent plane in two different directions. (The classic example is a saddle, which can be found on your body in the space between your thumb and forefinger, or along the inside of your neck.)
- Zero Curvature: A flat and infinite 2D plane
Below is a Representation of the Euclidean Surface
Based on the Definitions Above
What type of curvature does the Euclidean surface have?
Explain your thinking. Discuss your thoughts with others if possible.
Below is a Representation of the Elliptical/Spherical Surface
Based on the Definitions Above
What type of curvature does the Elliptical/Spherical surface have?
Explain your thinking. Discuss your thoughts with others if possible.
Below is a Representation of a Hyperbolic Saddle Surface
Based on the Definitions Above
What type of curvature does the Hyperbolic surface have?
Explain your thinking. Discuss your thoughts with others if possible.