Plotting lines and planes in 3D
Source: Many of the ideas and strategies in this worksheet are taken from the excellent resource on doing multivariate calculus in Sage by Prof. Sang-gu Lee, Mathematics Department, Sungkyunkwan University, Korea. The multivariate part of their Sage woksheets can be found at http://matrix.skku.ac.kr/Cal-Book/part2/part2.html. In this module we will learn how to plot lines, planes and surfaces in 3D. The three basic sage functions we will use most frequently for 3D plotting are:- plot3d()
- implicit_plot3d()
- parametric_plot()
-
plot3d?
implicit_plot3d?
parametric_plot?
This module assumes you know how to define and work with vectors. If not, please review/complete that module first.
Example 1: Find and plot each of the following as instructed:
- The line passing through the point and
parallel to the vector .
Show the line and the parallel vector in your plot.
- The line passing through the points and
- The plane passing through the point and perpendicular to the vector . Show the vector and the plane in your plot.
- The line of intersection of the planes and . Show the planes and the line of intersection in your plot.
(t + 2, t, -t + 1)
3D rendering not yet implemented
(-2*t + 2, 2*t, -t + 1)
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The plane is: -x - y - z + 3 = 0
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x = -r22 + 1 , y = r22 , z = r22
3D rendering not yet implemented
Exercise 1:
Do these problems from Sec.9.5 of our textbook:
5, 23, 45.
In all cases, use Sage to do your computations and to
graph your results.
Plotting other simple surfaces in 3D
We conclude with a brief glimpse of plotting more general surfaces in 3D.
Example 2: Plot each of the following as instructed:
- The equation in , and then in (on separate plots, please!).
- Graph the surface and describe what it looks like
- Graph the surface .
- show graphically that the curve of intersection of the surfaces and lies in a plane.
3D rendering not yet implemented
3D rendering not yet implemented
Example 2b: continued
By rotating the 3D plot above, it looks like an elliptic
cylinder, with center along the -axis.
3D rendering not yet implemented
3D rendering not yet implemented
Exercise 2:
Do these problems from Sec.9.6 of our textbook:
15(f), 23, 31.
Here is an extra exercise: Graph the following surface and
determine what it looks like
,
on , with red color.