Vectors and the Geometry of Space: Cylinders and Quadric Surfaces

about 2 months ago (Edited)

In this video I explore the wonderful world of cylinders and quadric surfaces. Cylinders are defined as a 3D surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve. Quadric surfaces are the graphs of a second-degree equation in three variables x, y, and z. Unlike quadric surfaces, cylinders have 1 of the x, y, or z variables missing from its equation. This makes the cylinder extend outwards along the missing variable. I also go over some exercises and show that the Earth can be more accurately modeled as an ellipsoid, which is a type of quadric surface. This section is part of the Vectors and Geometry of Space chapter in my Calculus book.

The topics covered as well as their timestamps are listed below.

Intro: 0:00
Calculus Book Reference: 0:50
Sections in Calculus Book Chapter: 1:03
Topics to Cover: 1:49

Cylinders and Quadric Surfaces: 3:22
Cylinders: 5:10
- Example 1: Parabolic Cylinder: 5:28
- Example 2: Circular Cylinders: 10:31
  - (a) Vertical Cylinder: 10:44
  - (b) Horizontal Cylinder: 14:45
- Note: 3D Surfaces vs 2D Traces: 17:40
Quadric Surfaces: 18:54
- Example 3: Ellipsoid: 24:23
- Example 4: Elliptic Paraboloid: 37:25
- Example 5: Hyperbolic Paraboloid: 45:01
- Example 6: Hyperboloid of One Sheet: 1:06:03
- Computer-Drawn Graphs of Quadric Surfaces: 1:12:19
  - Table 1: Graphs of Quadric Surfaces: 1:12:57
- Example 7: Hyperboloid of Two Sheets: 1:18:13
- Example 8: Elliptic Paraboloid: 1:29:36
- Applications of Quadric Surfaces: 1:41:25
Exercises
- Exercise 1: Modeling Earth as an Ellipsoid: 1:46:54
  - (a) Earth's Ellipsoid Equation: 1:48:06
  - (b) Parallels or Circular Lines of Equal Latitude: 1:53:24
  - (c) Meridians or Elliptical Lines of Equal Longitude: 1:58:56
- Exercise 2: Hyperbolic Paraboloid Generated from Lines: 2:05:51