Many families of shapes can be organized into a geometric object, where each point in the geometric object corresponds to a particular shape. For example, the family of all circles in the plane can be described by:
- Picking a point in the plane to be the center
- Picking a positive number to be the radius
We can combine this information together, and say that each specific circle corresponds to a point in $3$-space with positive $z$-coordinate.
Alternatively, we could use all of $3$-space rather if we let each point $(x,y,z)$ correspond to the circle with center $(x,y)$ and radius $e^z$.
In algebraic geometry, this notion is called a "moduli space". I don't think I've ever heard it given a name outside of algebraic geometry.
We could generalize a bit, and allow multiple points in the geometric object to correspond to the same shape. Then, we could represent a circle by a point in $(x,y,z,w)$ $4$-space such that $(x,y) \neq (z,w)$; the circle represented is the one with center $(x,y)$ and passing through $(z,w)$.
Of course, a point in $4$-space can be expressed as a pair of points in $2$-space, and they could be thought of as control points.
The use of control points really is mainly for the purposes of visualization, to let a person look at a shape and understand how moving the control points around would change the shape.