Help needed - Equation of a wedge shaped plane

[stu807]

Help needed - Equation of a wedge shaped plane!

Hi All,

I need some help to derive a formula to describe a plane. This I know already but this has all four sides with a different gradient.

Is there a formula I can use which I apply the X (lenght) any Y (width) coordinates and it will give me the Z?

I have attached a pic of the plane with the gradients on each side.

TIA

Stu

 

[HallsofIvy]

What do you mean by a "wedge shaped plane"? A plane cannot be "wedge shaped". Perhaps you mean a surface formed by four separate planes. In that case you will probably need a "piecewise" formula.

 

[stu807]

Ok, perhaps 'wedge' does not decribe it properly.

It is a 4 sided surface in 3D space where all four sides are at a different angle in relation to the base.

I hope the diagram shows this a little more clearly.

 

[ObsessiveMathsFreak]

Normally, lines, planes, etc are all describe using equations. E.g. x+2*y+z=4. Note the equality sign.

But if you want to do something like specifying half planes or enclosed regions, then you must describe it using inequalities. E.g. x+2*y+z<4

The best example of this is the difference between a circle and a disc. A circle of radius r is defined by x^2+y^2=r^2. But a disc of radius r, i.e., the area enclosed by the circle, is defined by x^2+y^2<r^2.

That's an enclosed space specified by only one equation, but if you wanted something like a closed off square, you would need several inequalities. For a square of side 2 centered and the origin, you need two inequalities. -1<x<1 and -1<y<1. (OK you could use the taxi cab mean, but let's not)

I still of course haven't answered your original question, namely, how to specify the three dimensional area. Now you need to mix and match equalities and inequalities.

Say your plane was described by x+2*y+z=4. If you combine this equality, with some inequalities, you can describe a region of this plane by giving some inequalities. Say for example -1<3x+y<2 and 3<z+2*x<5. You can then find the four corners of the wedge by solving the sets of equations that represent the boundaries of the inequality regions. i.e.

x+2y+z=4
3x+y=-1
z+2x=3

x+2y+z=4
3x+y=-1
z+2x=5

x+2y+z=4
3x+y=2
z+2x=3

x+2y+z=4
3x+y=2
z+2x=5

Edit: One word of warning. The equations of the boundaries must of course be independent for you to be able to solve for the corner points. So if you like your inequalities and equality must be independent as well. Whatever that means for inequalities.

Hope that helps

 

[stu807]

Cool, thanks!