Coordinates of intersection between an oblique rectangular pyramid and an angled plane

[andjo]

Hi All,

I'm trying to source a set of equations to calculate the x,y,z of where an angled plane intersects an oblique rectangular pyramid.

The pyramid has the following coordinates:
Centre of oblique rectangular pyramid: 70,35,350
Corner #1: 100,50,350
Corner #2: 100,-50,350
Corner #3: -100,-50,350
Corner #4: -100,50,350
Apex: 0,0,0

The angled plane is hinged at 0,0,175 and can tilt and roll. Assuming it's boundaries extend out well beyond the pyramid, it's dimensions are irrelavent.

Firstly, how can I calculate the intersecting coordinates (x,y,z) of the two shapes if the angled plane only tilts (for example to an angle of 45 degrees)?

Secondly, how can I calculate the intersecting coordinates (x,y,z) of the two shapes if the the angled can both tilt (45 degrees) and roll (for example 10 degrees)?

I've attached two example images to help better visualise what I'm trying to achieve.

Please could I request that anybody able to help not only provide the formula for this but also apply it using the able values so I can be sure to understand it.

A massive thanks in advance to anybody able to assist.

Regards
ANDJO

 

[blamocur]

Cool drawings!

But: "...request that anybody able to help not only provide the formula for this but also apply it using the able values..." -- you need to read the rules of this site (https://www.freemathhelp.com/forum/threads/read-before-posting.109846/) which, among other things says: Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck.

Some of the questions which come to mind:

1. When you talk of the tilt, in which direction do you tilt the plane. Do you always tilt in the same direction (which one?), and then rotate it around Z axis to do the roll?
2. Are you looking for the coordinates of intersections of the plane with non-horizontal edges of the pyramid ? With each of them?
3. Are you familiar with the equations of planes and straight lines in 3D?
4. More generally: where does this problem come from ? Homework? Practical task at work?

 

[andjo]

Hi blamocur,

Let me add some further context.

The pyramid is a a cone of light emitted from a source, the flat plane is a surface onto which the light is hitting.

Typically the surface the light is hitting would 45 degrees to the light as shown within image #1, however, this could alter by +/- 10 degrees in either direction (typically). I can actually solve this, although my approach is not elegant and I'm sure there is a better method.

In the second image the mirror as before could have anything from 35 - 55 degree angle of tilt and then the same on the other axis, i.e. +/- 10 degrees roll.

Tilt and roll share the same pivot point.

The x,y,z coordinates I'm looking at calculate are the four corners where the light would be hitting the intersecting plane. With only a single tilt then I believe two edges will be horizontal or vertical and the other will the angled. i.e. I would have an isosceles trapeizium. In the more complicated request I believe I would end up with a scalene trapezium.

My understnding of math is reasonable, however, I'm not that familiar with equations of planes and lines in 3D space, this is more because I've never had to use them though, not because I don't understand higher level math, just not applied certain principles for +/- 20yrs post high school.

The problem is a work place, practical problem.

 

[blamocur]

Tilting and "rolling" your plane is the same as applying these transforms to the plane's normal, so you can use point-normal form of the plane representation (https://en.wikipedia.org/wiki/Plane...m_and_general_form_of_the_equation_of_a_plane).
Then you can use a parametric representation of each pyramid's edge. Since they all pass through the origin you can represent it as [imath]t\bar p[/imath] where [imath]\bar p[/imath] is some point on the edge and [imath]t[/imath] is the parameter. For example, the first edge consists of points with coordinates (100t, 50t, 350t)
If you plug in the parametric representation of the edge into the equation of the plane you will get a simple linear equation for [imath]t[/imath]; when you solve it you can get the edge point which belongs to the plane. Then repeat this for the remaining three edges and you will get all points of the quadrangle on the cutting plane.
Good luck, and let me know if you have more questions.

 

[andjo]

Thanks, managed to apply this and convert to an excel tool over the weekend, works brilliantly