Planes
What is the definition of a plane?
A plane is a flat surface that extends in all directions. Can be defined by 3 non-colinear (not on a line) points or by a point and a vector (which is
perpendicular to the plane).
What is the equation for a plane?
The general equation for a plane is:
Ax + By + Cz + D = 0
An alternate form of this equation is:
A'x + B'y + C'z + D' = 0
A'= A/d, B'= B/d, C'= C/d, D'= D/d
where d = sqrt(A^2 + B^2 + C^2)
How can I find the normal vector to a plane?
Given three points (P1, P2, P3) the normal vector N is given by:
N = [P2 - P1] X [P3 - P1]
where the operation "X" is the cross product.
Given the equation for a plane:
Ax + By + Cz + D = 0
the vector [A B C] is normal to the plane.
How can I find the equation for a plane?
Given three points: P1, P2, P3
- Find the normal vector, N = [A B C].
- Find the coefficient D, by taking the dot product of N with any point P in the plane and negating the result:
N . P = -D
- Use the coefficients found (A,B,C,D) in the general equation for a plane.
How can I find the intersection of two planes?
The intersection of two non-parallel planes is a line.
The equation for this line can be found by solving the simultaneous equations of the planes. This can be done using a general method such as Gaussian elimination.
How can I find the intersection of a plane and a line?
The intersection of a plane and a line that are not parallel is a point. This
point can be found by solving the simultaneous equations for the plane and the
line using a general method such as Gaussian elimination.
How can I determine if a given point is in a particular plane?
Given a point P = (x,y,z), and a plane with normal N and point P1:
N . [P - P1] = 0
if P is in the plane.
How can I determine if a given line is in a particular plane?
Given two points on the line, P1 and P2:
If both points are in the plane, the line is in the plane.
Given a line defined by vector V and point P:
If P is in the plane and V is perpendicular to N (the normal to the
plane), the line is in the plane.
What is a half-plane?
A line which lies in a plane divides that plane into two half-planes.
A half-plane expresses an inequality. For example, for the equation x + y > 2, all the points in the x-y plane that satisfy the inequality are in one half-plane and those that do not are in the other. The dividing line is the equation x + y = 2.