Traces of a Surface: Review and Example on an Ellipsoid whose Traces are Ellipses

In this video I review traces of a surface, which are curves of intersection of a surface with planes parallel to the coordinate (x, y, z) planes. They are obtained by setting x, y, or z to a constant k and then obtaining a 2D equation. They are useful for determining the shape of a 3D surface, especially for basic surfaces whose cross sections are simple conic sections. I illustrate this with an ellipsoid, which has vertical and horizontal traces being ellipses.

Timestamps:

- Question 18: Traces of a Surface: 0:00
- Solution: Traces are curves of intersection of a surface with planes parallel to coordinate planes: 0:08
- Ellipsoid has vertical and horizontal ellipses as traces: 0:25
- Traces of ellipsoid obtained by setting x, y, z to a constant k separately: 1:24
- Setting z = k obtains a horizontal ellipse: 1:40
- Recap on parabolas, ellipses, hyperbolas, and circles: 2:58
- Setting x = k obtains a vertical ellipse: 3:49
- Setting y = k obtains a vertical ellipse: 5:10
- Right side of trace equation can't equal to zero for an ellipse, get trivial case at endpoints of surface: 5:42

Full video and playlists:

- Full video: https://youtu.be/VMKb69uDVt0
- HIVE notes: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-review
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EMiATrjv8HH3fPwKVsgU6P
- Vectors and the Geometry of Space playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .

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