Planes

Return to Course Outline

Previous Lecture

Anything You Want to Talk About?

Can you meet with me to ask questions about a problem set and then later to grade it? Absolutely!

Colloquium

The first math colloquium (a talk, often but not always by a mathematician visiting the department) of the semester is next Tuesday, the 20th.

It’s by Lisa Smith, of Wilmot Cancer Center, on “How to Leverage Your STEM Education to Succeed in Healthcare IT.”

(That’s the official title, I get the impression that a lot of the talk answers the question “what can I do with a math degree” (or maybe a degree in other STEM fields) with careers in healthcare IT.)

4:30 - 5:20

Newton 204.

Planes

Based on “Equations for a Plane” in section 1.5.

Questions?

Given a point in a plane and a line perpendicular to it, how do you find the plane’s equation? For example…

The key insight is that in the scalar equation for the plane, 3 of the 4 coefficients are the components of a vector normal (perpendicular) to the plane:

In the case where you know a point in the plane and a perpendicular line, the line’s direction vector gives you the normal, and then you can plug the point into the resulting plane equation and solve for the 4th coefficient:

Key Ideas

The vector equation for a plane with normal n and points P and Q in the plane: n • PQ = 0

The scalar equation for a plane: Ax + By + Cz + D = 0, or Ax + By + Cz = -D

The distance from plane Ax + By + Cz + D = 0 containing point Q to point P:

Warm-Up Examples

Find an equation for the plane perpendicular to ⟨3, -1, 1⟩ and containing point (2,0,4).

Much like the point-and-line example above, you can plug the components of the perpendicular vector into the scalar equation for the plane, and then use the point to solve for D:

How about an equation for the plane containing the origin, (1,0,0), and (0,0,1)?

In general, you can find an equation for a plane given 3 points by using the points to calculate vectors parallel to the plane, using the cross product of those vectors as the normal, and solving for D. In this case the natural vectors to use are (1,0,0) - (0,0,0) = ⟨1,0,0⟩ and (0,0,1) - (0,0,0) = ⟨0,0,1⟩.

This problem is simple enough that you can check less formally that this method produces reasonable results: since all 3 points lie along the x and/or z axes, a vector along the y axis (such as ⟨0,1,0⟩ produced by our cross product) is intuitively a good normal. Furthermore, the plane is just the xz-plane, which does indeed have equation y = 0.

Applications

One of my research interests is computer graphics and crystal aggregates — whether there are mathematical models that can describe, and algorithms that can then create pictures of, realistic-looking clusters of crystals. Because crystals have planar faces, and cut each other off as they grow, one of the things my programs do a lot of is figuring out how planes interact with each other, with lines, points, etc.

See if you can figure out the mathematics of doing the following. For each method you figure out, also see if you can come up with an example that illustrates it. You can assume that…

• Planes are represented by scalar equations of the form Ax + By + Cz = D
• Line segments are represented by their endpoints
• Points are represented by their x, y, and z coordinates.

Determining whether a point lies in a plane

Plug the point into the plane’s scalar equation. If the result is not 0 then the point isn’t on the plane, if the result is 0 the point is on the plane.

Determining whether a line segment lies in a plane

Use the above to see if both endpoints lie in the plane

Determining where a line (not necessarily just a line segment, but identified by one that it contains) intersects a plane

???

Determining whether a line segment intersects a plane at all, and if so where.

???

Determining whether 2 points lie on the same side of a plane

???

Determining whether a point lies inside a convex polyhedron (hint: assume the planes containing the faces of the polyhedron all have their normals pointing out of the polyhedron)

???

Problem Set

Problem set 3, on lines and planes, is available.

Work on it through Thursday of next week; grade it through Thursday of the following week.

See the handout for more information.

Next

Talk about more of the application examples, maybe develop and look at concrete examples of some of the solutions.

No new reading, but keep thinking about the problems and come to class ready to talk, or ask questions, about them.

Next Lecture