Upper Division Mathematics: Computational Differential Geometry

Roger Williams University Course Catalog Listing: DMAT 451 - Computational Differential Geometry

DMAT 451 - Computational Differential Geometry - Learning Outcomes

• 1. To use computer graphing tools to visualize 2D and 3D curves and surfaces
• 2. To understand and compute various metrics about parametric and non-parametric curves and surfaces
• 3. To understand and compute the key concept of curvature, and understand its relationship to derivatives and differential equations
• 4. To understand the role of motions in geometry
• 5. To understand and compute the Frenet frames for curves
• 6. To understand and compute the concept of the derivative for vector fields
• 7. To understand and compute the local Gauss map for surfaces
• 8. To understand and compute the concept of orientability of surfaces
• 9. To understand and compute Gaussian and Mean curvature
• 10. To understand and compute Ruled and Minimal Surfaces
• 11. To be introduced to the Gauss-Bonnet theorem

DMAT 451 - Computational Differential Geometry - Syllabus of Topics

1.	Getting Started
	1.1	Email and Chat
	1.2	Learning About the Course
	1.3	Required Hardware
	1.4	Software Fundamentals

2.	Curves
	2.1	Euclidean Spaces
	2.2	2D and 3D Parametric Curves
	2.3	Arclength
	2.4	Curvature vs. Derivative
	2.5	Angles
	2.6	Catalog of Famous Curves

3.	Alternative Ways of Plotting Curves
	3.1	Implicit Curves
	3.2	Contour Plots
	3.3	Polar Coordinates
	3.4	New Curves from Old

4.	Solving Curvature Equations
	4.1	Euclidean Motions
	4.2	Intrinsic Equations
	4.3	Assigned Curvature

5.	Global Properties of Plane Curves
	5.1	Total Signed Curvature
	5.2	Turning Numbers
	5.3	Rotation Index
	5.4	Convexity
	5.5	Constant Width
	5.6	Support Functions

6.	Space Curves
	6.1	Tangent, Normal, Binormal Frames
	6.2	Curvature and Torsion
	6.3	Frenet Formulas
	6.4	Arbitrary Speed Curves
	6.5	Tubes and Tori

7.	Fundamental Theorem of Space Curves
	7.1	Assigned Curvature and Torsion
	7.2	Contact
	7.3	Curves That Lie on a Sphere
	7.4	Curves of Constant Slope

8.	Calculus of Euclidean Space
	8.1	Tangent Vectors and Directional Derivatives
	8.2	Tangential Maps
	8.3	Vector Fields
	8.4	Derivatives of Vector Fields

9.	Surfaces in Euclidean Space
	9.1	Patches
	9.2	Local Gauss Map
	9.3	Regular Surfaces
	9.4	Level Surfaces
	9.5	Catalog of Famous Surfaces	

10.	Non-Orientable Surfaces
	10.1	Orientability
	10.2	Mobius Strip and Klein Bottle
	10.3	Projective Planes

11.	Surface Metrics
	11.1	Distance
	11.2	Isometries
	11.3	Conformal Maps
	11.4	Gaussian and Mean Curvature
	11.5	Non-Parametrically-Defined Surfaces

12.	Ruled and Other Surfaces
	12.1	Examples
	12.2	Curvature
	12.3	Surfaces of Revolution
	12.4	Examples of Minimal Surfaces

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