1. ## Planes

Help! I'm not very mathy, I'm doing a programming course at University, and this semester I have a Maths and Physics for Games subject. My lecturers aren't very helpful, we ask for examples and they get mad, we ask them to provide solutions for us to check our answers against and they get mad. So yeah, with a teaching strategy akin to Profesor Umbridge I'm pretty lost.

Anyway, I desperately need to understand how to use the generalised and parameterised plane equations, so that I can complete this assignment by sunday and not fail.

I understand the parameterized one a little bit; a plane is an origin point plus two arbitrarily scaled vectors, the only question on planes I've been able to answer for my assignment was to develop the plane equation from three given points...
P(s,t) = P0+s(P1-P0)+t(P1-P0)
= P0 +su+tv

Mainly I need someone to help me out with the generalised equation because I don't understand what anything represents and therefore how to find out other things from it, like if two planes are parallel, and how to find the perpendicular vector etc.
So any examples or explanations on...

0=ax+by+cz-(ax0+by0+cz0)
0=ax+by+cz+d

...you can give me would be greatly appreciated!!

2. Originally Posted by balooneybob
Help! I'm not very mathy, I'm doing a programming course at University, and this semester I have a Maths and Physics for Games subject. My lecturers aren't very helpful, we ask for examples and they get mad, we ask them to provide solutions for us to check our answers against and they get mad. So yeah, with a teaching strategy akin to Profesor Umbridge I'm pretty lost.

Anyway, I desperately need to understand how to use the generalised and parameterised plane equations, so that I can complete this assignment by sunday and not fail.

I understand the parameterized one a little bit; a plane is an origin point plus two arbitrarily scaled vectors, the only question on planes I've been able to answer for my assignment was to develop the plane equation from three given points...
P(s,t) = P0+s(P1-P0)+t(P1-P0)
= P0 +su+tv

Mainly I need someone to help me out with the generalised equation because I don't understand what anything represents and therefore how to find out other things from it, like if two planes are parallel, and how to find the perpendicular vector etc.
So any examples or explanations on...

0=ax+by+cz-(ax0+by0+cz0)
0=ax+by+cz+d

...you can give me would be greatly appreciated!!
1. Let me start with an example: Consider the "roof" of an umbrella whose position in 3d is determined by the direction of the handle and one point on the roof. The handle should be perpendicular to the roof of the umbrella . An other expression for perpendicular is normal. So the handle describes the normal vector of the roof of the umbrella.

2. You should have learned that the scalar product of 2 vectors equals zero if the 2 vectors are perpendicular to each other. Take the ribs of the umbrella's roof as a vector pointing at a point of the roof. This point has the coordinates P(x, y, z) that means we are talking about an arbitrary point of the plane. The fixed point $\displaystyle P_0(x_0, y_0, z_0)$ is the point of intersection between the handle and the roof.
Then the vector $\displaystyle \overrightarrow{P_0P} = (x,y,z) - (x_0, y_0, z_0)$ must lie completely in the plane.

3. The normal vector of a plane is $\displaystyle \vec n = (a,b,c)$. According to 2.) the product of $\displaystyle \vec n \text{ and } \overrightarrow{P_0P}$ must equal zero:

$\displaystyle (a,b,c) \cdot \left((x,y,z) - (x_0, y_0, z_0) \right) = 0$

yields:

$\displaystyle ax+by+cz-ax_0-by_0-cz_0=0$

$\displaystyle ax+by+cz+d=0$

This equation is valid for every point in the plane thus this equation describes the plane completely.

Sorry if these "explanations" are too simple for you.

3. Simple is what I needed! Thank you I'm sure this will help heaps. I'm still not 100% sure on how to work backwards from the equation to determine other things... If you've got a plane x+y+z=0, would the normal be a vector (1,1,1)? And if d=0 is does that mean that P0 is the origin or something?

4. Originally Posted by balooneybob
Simple is what I needed! Thank you I'm sure this will help heaps. I'm still not 100% sure on how to work backwards from the equation to determine other things... If you've got a plane x+y+z=0, would the normal be a vector (1,1,1)? Yes
And if d=0 is does that mean that P0 is the origin? Yes

5. I have to find any point in the plane, that isn't the origin. I tried subbing stuff into the equation but that didn't seem to work - is it just me doing this in a retarded way or is it the wrong approach? I've got an equation where d=0, so essentially all i have to work with is the normal vector, right? because (0,0,0) isn't helpful that I can think of at least... *is confused*

6. A plane is determined by two numbers. Substitute any two numbers for, say, x, and y, and solve for z. If, for example, 3x+ 2y- 6z= 0, then choose, for no particular reason, x= 1 and y= 2. 3x+ 2y= 3+ 4= 7 so your equation becomes 7- 6z= 0. Then 6z= 7 and z= 7/6. A point on the plane is (1, 2, 7/6).

7. Thanks! I was trying to do that but must have done it in some retarded way and got stressed out, wondering if it was the right way to go about it anyway! Thanks so much for your help guys, I really appreciate it!