Plane transformations

Planes can be defined with different forms such as the parametric form, cartesian form or normal form. We show how we can transform between these representations of the same plane.

Transform a cartesian plane form to the normal form

We have a plane in the cartesian form equation and want to transform it to the normal form equation. For this we need to find the vectors equation and equation. The vector equation is the normal vector (it points out of the plane and is perpendicular to it) and is obtained from the cartesian form from equation, equation and equation: equation. Now we need to find equation which is a point on the plane. There are infinitely many points we could pick and we just need to find any one solution for equation, equation, and equation. If we set all but equation and equation to zero and divide by equation, we get the point equation. Now we can represent the plane in normal form:

(1)   equation

Transform a normal plane form to the cartesian form

We are given a plane in the normal form equation and want to transform it to the cartesian form equation. This can be achieved by simply expanding the normal form:

(2)   equation

With equation with equation, equation, equation and equation.

Transform a parametric plane form to the cartesian form

We are given a plane in the parametric form equation and want to transform it to the cartesian form equation. First we need to calculate the normal vector equation of the plane by using the cross product:

    equation

We calculate equation as equation and equation, equation and equation are the components of the equation vector:

(3)   equation

Transform a cartesian plane form to the parametric form

We are given a plane in cartesian form equation and want to transform it to the parametric form equation. We solve the cartesian form for one variable, for example equation:

    equation

We now define equation and equation as equation and equation and get the parametric form as:

    equation

Transform a normal plane form to the parametric form

We are given a plane in the normal form equation and want to transform it to the parametric form equation. You can go the detour of transforming to the cartesian form first as shown above which is easy because the transformation between normal and cartesian form is easy. If you want to transform them directly, there are two ways: you can find two points on the plane (values which solve the normal form equation) and use them to span the plane (vectors equation and equation). Alternatively, you can create a set of linear equations from the following conditions:

  • The normal vector equation is perpendicular to the vectors equation and equation of the parametric form. Using the dot product, this can be expressed as equation
  • The two vectors equation and equation must be linear independent: equation

Transform a parametric plane form to the normal form

This transformation is nearly identical to the transformation from the parametric form to the cartesian form. We are given a plane in the parametric form equation and want to transform it to the normal form equation. We only need to calculate the normal vector equation of the plane by using the cross product and then we have all information for the normal form:

    equation