Objectives
- To understand graphs of fundamental planes
- To understand how intersections of graphs with fundamental planes corresponds to algebraically substituting a coordinate value
Section 3.2 Graphs in 3D
Subsection 3.2.1 Fundamental Planes in 3D
This lesson introduces the term fundamental plane as the plane defined by having a rectangular coordinate at a fixed value, i.e. \(x\) or \(y\) or \(z\) equal to a constant. The intersection of fundamental planes is discussed and then more complicated surfaces are explored by looking at the curve created by intersecting with a fundamental plane.
Activity 3.2.1. Intersections of Fundamental Planes.
(a)
Draw a graph in three-dimensional space for each description and give a symbolic/algebraic description for the set of points described:
- the intersection of hte plane \(x=1\) with the plane \(y=2\) (the intersection consists ONLY of the points that are on both planes at the same time; draw ONLY those points, that is, do not draw the planes on your final plot)
- the intersection of hte plane \(y=-1\) with the plane \(z=4\)
Hint.
Look at the first several examples in the CalcVR lesson to see what kind of shapes you should get for this problem.
Subsection 3.2.2 More Fundamental Planes and Graphs
The second lesson on Fundamental Planes and Graphs in 3D continues with questions and examples for how different fundamental planes intersect with surfaces. The key point to understand is that this intersection is the geometric analog of substituting the variable value given in the equaiton for the fundamental plane. This lesson leads to studying the basic surface types of Cylinder Surfaces and Quadric surfaces.
Activity 3.2.2.
Consider \(S=\{(x,y,z)\in \mathbb{R}|z=x^2+y^2\}\) and \(z=4\text{.}\)The “intersection” of the plane and the set \(S\) consists of the points that are on the plane and that are also in the set \(S\text{.}\) Be sure to follow all the instructions from (a) to (c).
(a)
Draw the intersection on a Cartesian plane, identifying and labeling both axes. Note: to draw the intersection one does not need to draw or even know how the graph of set \(S\) looks.
(b)
Draw the intersection in three-dimensional space. Make sure that all the points on the graph are on the corresponding fundamental plane.
(c)
Find the coordintates of three of the points on the intersection.
Activity 3.2.3.
Draw each of the following sets in Cartesian three-dimensional space. To do so one doesn’t need to draw or know how the graph of \(z=xy^2\) looks. Each one of the sets is the intersection of a plane with the surface that is the graph of \(z=xy^2\text{.}\)
(a)
\(\{(x,y,z)\in\mathbb{R}|z=xy^2 \text{ and } x=2\}\)
(b)
\(\{(x,y,z)\in\mathbb{R}|z=xy^2 \text{ and } z=1\}\)
Activity 3.2.4.
Draw each of the following sets in Cartesian three-dimensional space. To do so one doesn’t need to draw or know how the graph of \(z=x \sin(y)\) looks. Each one of the sets is the intersection of a plane with the surface that is the graph of \(z=x \sin(y)\text{.}\)
(a)
\(\{(x,y,z)\in\mathbb{R}|z=x \sin(y) \text{ and } y=0\}\)
(b)
\(\{(x,y,z)\in\mathbb{R}|z=x \sin(y) \text{ and } x=2\}\)
(c)
\(\{(x,y,z)\in\mathbb{R}|z=x \sin(y) \text{ and } z=1\}\)
