Realistic point-plane distance problem?

[nomadreid]

I am teaching the standard 3-D stuff, including the distance between a given point and a given plane. Yet searching for a good example (or test question) in the context of a real situation, I find on the Internet only the abstract "a point is... a plane is... find the distance". The only realistic examples I found for distance in 3-D are only finding the distance between two given points. Can anyone do better? Thanks. (Thanks also, presumably, from my students.)

 

[andrewkirk]

Perhaps something involving capacitance?
Say we place a small metal sphere carrying net charge q coulombs at coordinates (a,b,c) and a flat sheet of metal in the plane dx+ey+fz=g, what is the maximum electric field at the surface of the metal sheet, prior to any rearrangement of charges on the sheet? That then leads on to follow-up questions about the pattern formed by the rearrangement of charges on the metal sheet.

Also, something from engineering: what is the shortest strut that can be used to support a sloping roof with a given plane equation from a given support point.

 

[nomadreid]

AndrewKirk, many thanks. Both suggestions are very good. (How far I would want to follow up on the capacitance question is another question, since it is a maths class, where the physics background of the students is varied.) I'll use them.

 

[tygerdawg]

An example from advanced programming of industrial 6-axis robots.

• All points are 6DOF points defined in the robot base frame coordinate system. For example: pointB = {x, y, z, yaw, pitch, roll} or perhaps = {x, y, z, rotation, rotationy, rotationz}. All depends on how one defines the orientation component of a 6DOF point.
• A work surface has a relative coordinate system assigned to it by the robot using typical program language function. Requires points on the work surface frame coordinate system: point on frame Origin (pntOrigin), point on frame X Axis (pntXAxis), point in the positive XY plane of frame (pntYAxis). Work surface frame definition calculated using typical robot program function frmWorkSurface = FRAME(pntOrigin, pntXAxis, pntYAxis, pntOrigin). This produces a work surface frame transformation frmWorkSurface = {frmWSX, frmWSY, frmWSZ, frmRotX, frmRotY, frmRotZ}
• Robot end effector Tool Center Point (TCP) is defined as working tool point where action occurs. Defined in space pntHERE = {pntHereX, pntHereY, pntHereZ, pntHereRotX, pntHereRotY, pntHereRotZ}.
• Problem is to know the distance from the TCP to the (a) origin of the work surface frame...which is easy to calculate, and (b) perpendicular distance from the TCP to the plane...which is a bit more involved.
• Reason: the robot process may be something like dispensing or welding where the TCP-to-surface distance is a Critical-To-Quality (CTQ) factor affecting final product quality.

 

[nomadreid]

tygerdawg, many thanks. Excellent example, and very realistic. I have students interested in robotics, so as soon as I phrase this according to their understanding, it should be a big hit!