What is a "control point"?

Question

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) )

There seems to be a bias towards ascribing a computer-graphics/CAD basis/usage for the term, but is it really limited to that field? I was thinking that control point might be a term used more generally in mathematics/geometry/topology to describe sets of points that describe curves/surfaces/manifolds -- is it a mistake to think it a general term?

Answer 1

People in CAGD use the term "control point" in a fairly general way. Typically, we are dealing with polynomials. Suppose we are given a polynomial $C:\mathbf{R} \to V$ of degree $m$, where $V$ is some vector space, which is usually $\mathbf{R}^2$ or $\mathbf{R}^3$. The image $C([0,1])$ is then a "curve". Suppose we choose some basis $\phi_{0}, \ldots, \phi_m$ for the space of polynomials of degree $m$. Then we can find $P_0, \ldots, P_m \in V$ such that $$ C(t) = \sum_{i=0}^m \phi_i(t)P_i $$ Then $P_0, \ldots, P_m$ are called the "control points" of the curve with respect to the chosen basis.

Depending on which basis you choose, the corresponding control points may or may not provide a useful way of adjusting (i.e. controlling) the shape of the curve. For a basis to be useful in this way, it almost always will need to form a partition of unity. The Bernstein basis is the most commonly used one; in this case, the control points are called "Bezier" control points. Bases consisting of Lagrange polynomials are also fairly common, but the corresponding control points don't have any special name.

The extension to surfaces and volumes is straightforward. For tensor-product surfaces, you get two-dimensional arrays of control points.

So, in short, a set of "control points" is just the set of coefficients used to represent some given (vector valued) polynomial in some chosen basis.

Answer 2

Let a manifold be parameterized by $t = (t_1, \dots, t_n) \in \Bbb{R}^n$ and additionally the points $p_{i_1, \dots, i_n}, \ i_j = 1\dots m$. Then the points $p_{\bar{i}}$ are the control points of the manifold. There. Making definitions can be easy when there's no desired properties.

Referring back to the wiki article, you could modify the above definition to make sure that the manifold always lies within the convex hull.

The term partitions of unity used in the article, I've seen defined in manifold and topology books, so their definition is pretty general in the sense that it uses other definitions that are for all practical purposes as general as they were required to be in general topology books.

Answer 3

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.