Cg Programming/Unity/Bézier Curves

This tutorial discusses one way to render quadratic Bézier curves and splines in Unity. This is the first of a few tutorials that demonstrate useful functionality of Unity that is not directly associated with shaders; thus, no shader programming is required and all the presented code is in C#.

There are many applications of smooth curves in computer graphics, in particular in animation and modeling. In a 3D game engine, one would usually use a tube-like mesh and deform it with vertex blending as discussed in instead of rendering curves; however, rendering curved lines instead of deformed meshes can offer a substantial performance advantage.



Linear Bézier Curves
The simplest Bézier curve is a linear Bézier curve $$B(t)$$ for $$t$$ from 0 to 1 between two points $$P_0$$ and $$P_1$$, which happens to be the same as linear interpolation between the two points:

$$B(t) = (1-t) P_0 + t P_1$$

You might be fancy and call $$1-t$$ and $$t$$ the Bernstein basis polynomials of degree 1, but it really is just linear interpolation.



Quadratic Bézier Curves
A more interesting Bézier curve is a quadratic Bézier curve $$B(t)$$ for $$t$$ from 0 to 1 between two points $$P_0$$ and $$P_2$$ but influenced by a third point $$P_1$$ in the middle. The definition is:

$$B(t) = (1-t)^2 P_0 + 2(1-t)t P_1 + t^2 P_2$$

This defines a smooth curve $$B(t)$$ with $$t \in [0,1]$$ that starts (for $$t=0$$) from position $$P_0$$ in the direction to $$P_1$$ but then bends to $$P_2$$ (and reaches it for $$t=1$$).

In practice, one usually samples the interval from 0 to 1 at sufficiently many points, e.g. $$B(0), B(0.05), B(0.1), B(0.15), B(0.2), ..., B(1)$$ and then renders straight lines between these sample points.

Curve Script
To implement such a curve in Unity, we can use the Unity component. Apart from setting some parameters, one should set the number of sample points on the curve with the function. Then the sample points have to be computed and set with the function. This is can be implemented this way:

Here we use an index  from 0 to   to count the sample points. From this index  a parameter   from 0 to 1 is computed. The next line computes $$B(t)$$, which is then set with the function.

The rest of the code just sets up the  component and defines public variables that can be used to define the control points and some rendering features of the curve.

To use this script create a C# Script in the Project Window and name it Bezier_Curve, double-click it, copy & paste the code above, save it, create a new empty game object (in the main menu: GameObject > Create Empty) and attach the script (drag the script from the Project Window over the empty game object in the Hierarchy Window).

Then create three more empty game objects (or any other game objects) with different(!) positions that will serve as control points. Select the game object with the script and drag the other game objects into the slots Start, Middle, and End in the Inspector. This should render a curve from the game object specified as “Start” to the game object specified as “End” bending towards “Middle”.



Quadratic Bézier Splines
A quadratic Bézier Spline is just a continuous, smooth curve that consist of segments that are quadratic Bézier curves. If the control points of the curves were chosen arbitrarily, the spline would neither be continuous nor smooth; thus, the control points have to be chosen in particular ways.

One common way is to use a certain set of user-specified control points (green circles in the figure) as the $$P_1$$ control points of the segments and to choose the center positions between two adjacent user-specified control points as the $$P_0$$ and $$P_2$$ control points (black rectangles in the figure). This actually guarantees that the spline is smooth (also in the mathematical sense that the tangent vector is continuous).

Spline Script
The following script implements this idea. For the j-th segment, it computes $$P_0$$ as the average of the j-th and (j+1)-th user-specified control points, $$P_1$$ is set to the (j+1)-th user-specified control point, and $$P_2$$ is the average of the (j+1)-th and (j+2)-th user-specified control points:

Each individual segment is then just computed as a quadratic Bézier curve. The only adjustment is that all but the last segment should not reach $$P_2$$. If they did, the first sample position of the next segment would be at the same position which would be visible in the rendering. The complete script is:

The script has to be named Bezier_Spline and works in the same way as the script for Bézier curves except that the user can specify an arbitrary number of control points. For closed splines, the last two user-specified control points should be the same as the first two control points. For open splines that actually reach the end points, the first and last control point should be specified twice.

Summary
In this tutorial, we have looked at:
 * the definition of linear and quadratic Bézier curves and quadratic Bézier splines
 * implementations of quadratic Bézier curves and quadratic Bézier splines with Unity's LineRenderer component.