Geometry
Analyze local and global properties of curves and surfaces with suitable representations and conditions.
Meaning
University-level geometry extends beyond plane and solid figures to study distances, curvature, symmetry, and structures defined by equations. It includes branches such as differential and algebraic geometry. The following is one example from surface geometry.
Example
Join the north pole to points on the equator at longitudes 0 and 90 degrees using shortest great-circle arcs. The spherical triangle has three 90-degree angles, totaling 270 degrees, unlike a plane triangle. The two meridians meet at the pole with the 90-degree longitude difference, and each meets the equator at a right angle. These sides are shortest great-circle arcs, so the plane parallel-line proof of a 180-degree angle sum does not apply. The 180-degree angle sum for plane triangles cannot be applied unchanged to every surface. Specify the space, distance, and lines being used.
Practice for this concept
Scope and references
Topics
Curves and surfaces, curvature, differential geometry, algebraic geometry
Learning goal
Analyze local and global properties of curves and surfaces with suitable representations and conditions.
Helpful background
Explore the coordinate geometry, linear algebra, calculus, or algebra needed for the chosen topic.
Curvature and tangents to implicit curves
These tasks do not assess surface curvature or general structures in differential or algebraic geometry.