This is a collection of animations that interactively demonstrate 2D and 3D parametric curves, the tangent and normal vectors and the osculating circle, and the parameterization of surfaces.
Type of Material:
classroom demos, student exploration
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Identify Major Learning Goals:
To visualize via animation and interactivity parameterization and unit tangent and normal vectors and the osculating circle.
Target Student Population:
Students in second and third semester calculus courses.
Prerequisite Knowledge or Skills:
A practical knowledge of parameterization and basic vector calculus.
Evaluation and Observation
This is a collection of 13 animations within a larger collection that are geared towards the visualization of parameterization and the tangent and normal vectors and the osculating circle. The animations are interactive in that the user can rotate the axes, start and stop the animation, zoom in and out, and move the cross-section at will. In the 2D animations, the student will get a strong sense of how the tangent vector changes in both direction and magnitude as the parameter changes. In the 2D and 3D curvature examples, the student can visually grasp how the unit tangent and normal vectors and the osculating circle change as the parameter changes. The parameterized surface animation allows the student to visualize the relationship between the rectangles in the uv-plane and the patches of the toroidal surface in xyz-space. The animations are all color-coded and the selection of curves and surfaces are specifically chosen to demonstrate the geometry of parameterization.
There is no explanation of the underlying concepts or links to pages that explain the underlying concepts. In the particles in motion animations, the parametric equations are not shown for three of the examples.
Potential Effectiveness as a Teaching Tool
This collection will serve as a powerful demonstration of the concepts by the instructor or an exploration session where the student can work in a computer lab. The entire demonstration of any one of these animation should take only a minute or two, allowing the instructor enough time for the rest of the lecture. After only a few minutes, the students will have clear idea of the relationship between parameterization and the accompanying geometry.
Ease of Use for Both Students and Faculty
Next to each animation there are easy to understand instructions on how to interact with the graph. Most of the interactivity just involves double-clicking or grabbing and dragging a special object. Students who read the instructions will have no trouble working with the animations.