## 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 and want to transform it to the normal form . For this we need to find the vectors and . The vector is the normal vector (it points out of the plane and is perpendicular to it) and is obtained from the cartesian form from , and : . Now we need to find 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 , , and . If we set all but and to zero and divide by , we get the point . Now we can represent the plane in normal form:

(1)

## Transform a normal plane form to the cartesian form

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

(2)

With with , , and .

## Transform a parametric plane form to the cartesian form

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

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

(3)

## Transform a cartesian plane form to the parametric form

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

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

## Transform a normal plane form to the parametric form

We are given a plane in the normal form and want to transform it to the parametric form . 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 and ). Alternatively, you can create a set of linear equations from the following conditions:

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

## 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 and want to transform it to the normal form . We only need to calculate the normal vector of the plane by using the cross product and then we have all information for the normal form: