I have a question about the notation of the plane equation (seen in day 402):
a, b, c are the components of the plane normal;
x, y, z are the components of the known point on the plane;
x0, y0, z0 are the point that we test to see if it's on the plane;

why do we say that the plane equation is

ax + by + cz + d = 0

where d is basically a "complex" term in the equation ( d = -ax0 - by0 - cy0 ) that needs to be developed to get a result from the equation.

ax + by + cz - ax0 - by0 - cy0 = 0

I (think I) understand the equations, and that they are the same, just no why we use the 'd' notation.

mrmixer

x, y, z are the components of the known point on the plane;
x0, y0, z0 are the point that we test to see if it's on the plane;

You've got this the wrong way around. (x,y,z) is the (test) point on the plane, and (x0, y0, z0) is the reference point. Any point (x,y,z) that satisfies

ax + by + cz + d = 0

is by definition on that plane. In a sense, the d parameter encodes the "height" of the plane (along the normal). By changing the reference point (x0, y0, z0), you change d.

The beginning of this Mathworld article might be illuminating: http://mathworld.wolfram.com/Plane.html.

Note the dot product definition. If the dot product is positive (angle less than 90 degrees), then the point (x, y, z) is "above" the plane, and ax + by + cz + d > 0. While if the dot product is negative (more than 90 degrees, then the test point is "below" the place, and ax + by + cz + d < 0.

Equivalently you could state this as:

ax + by + cz > -d, above plane
ax + by + cz < -d, below plane
ax + by + cz = -d, on plane

Thanks for the clarifications.
But why do we say that the definition is ax + by + cz + d = 0 with d = -ax0 -by0 -cy0 instead of directly saying it's the full equation ax + by + cz - ax0 - by0 - cy0 = 0 ? Are there cases where d would be something else ?

mrmixer
Thanks for the clarifications.
But why do we say that the definition is ax + by + cz + d = 0 with d = -ax0 -by0 -cy0 instead of directly saying it's the full equation ax + by + cz - ax0 - by0 - cy0 = 0 ? Are there cases where d would be something else ?

No but if x0 y0 and z0 are known it's better to simplify the equation and get rid of useless data. Especially when you will be using the coefficients in other formulas. It doesn't matter which point was the known point, there are an infinite amount of possible known points on the plane but all of them and up with the same d.

The original formula is really dot(n, p-p0) = 0 aka the line from the known point to the tested point is perpendicular to the normal.

this expands to a(x - x0) + b(y - y0) + c(z - y0) = 0 which after applying distributivity and commutativity ends up with ax + by + cz - ax0 - by0 - cy0 = 0.

ratchetfreak
No but if x0 y0 and z0 are known it's better to simplify the equation and get rid of useless data. Especially when you will be using the coefficients in other formulas.

Thanks, that make some sense.