Beziers, curves and paths¶
Bezier curves are a mathematical approximation of natural geometric shapes. We use them to represent a curve with as little information as possible and with a high level of flexibility.
Unlike more abstract mathematical concepts, Bezier curves were created for industrial design. They are a popular tool in the graphics software industry.
They rely on interpolation, which we saw in the previous article, combining multiple steps to create smooth curves. To better understand how Bezier curves work, let’s start from its simplest form: Quadratic Bezier.
Take three points, the minimum required for Quadratic Bezier to work:
To draw a curve between them, we first interpolate gradually over the two
vertices of each of the two segments formed by the three points, using values
ranging from 0 to 1. This gives us two points that move along the segments as we
change the value of
t from 0 to 1.
We then interpolate
q1 to obtain a single point
r that moves
along a curve.
This type of is called a Quadratic Bezier curve.
(Image credit: Wikipedia)
Building upon the previous example, we can get more control by interpolating between four points.
We first use a function with four parameters to take four points as an input,
We apply a linear interpolation to each couple of points to reduce them to three:
We then take our three points and reduce them to two:
And to one:
Here is the full function:
The result will be a smooth curve interpolating between all four points:
(Image credit: Wikipedia)
Cubic Bezier interpolation works the same in 3D, just use
Adding control points¶
Building upon Cubic Bezier, we can change the way two of the points work to
control the shape of our curve freely. Instead of having
p3, we will store them as:
point0 = p0: Is the first point, the source
control0 = p1 - p0: Is a vector relative to the first control point
control1 = p3 - p2: Is a vector relative to the second control point
point1 = p3: Is the second point, the destination
This way, we have two points and two control points which are relative vectors to the respective points. If you’ve used graphics or animation software before, this might look familiar:
This is how graphics software presents Bezier curves to the users, and how they work and look in Godot.
Curve2D, Curve3D, Path and Path2D¶
Using them, however, may not be completely obvious, so following is a description of the most common use cases for Bezier curves.
Just evaluating them may be an option, but in most cases it’s not very useful. The big drawback with Bezier curves is that if you traverse them at constant speed, from
t = 0 to
t = 1, the actual interpolation will not move at constant speed. The speed is also an interpolation between the distances between points
p3 and there is not a mathematically simple way to traverse the curve at constant speed.
Let’s do a simple example with the following pseudocode:
As you can see, the speed (in pixels per second) of the circle varies, even though
t is increased at constant speed. This makes beziers difficult to use for anything practical out of the box.
Drawing beziers (or objects based on the curve) is a very common use case, but it’s also not easy. For pretty much any case, Bezier curves need to be converted to some sort of segments. This is normally difficult, however, without creating a very high amount of them.
The reason is that some sections of a curve (specifically, corners) may require considerable amounts of points, while other sections may not:
Additionally, if both control points were
0, 0 (remember they are relative vectors), the Bezier curve would just be a straight line (so drawing a high amount of points would be wasteful).
Before drawing Bezier curves, tessellation is required. This is often done with a recursive or divide and conquer function that splits the curve until the curvature amount becomes less than a certain threshold.
The Curve classes provide this via the
Curve2D.tessellate() function (which receives optional
stages of recursion and angle
tolerance arguments). This way, drawing something based on a curve is easier.
The last common use case for the curves is to traverse them. Because of what was mentioned before regarding constant speed, this is also difficult.
To make this easier, the curves need to be baked into equidistant points. This way, they can be approximated with regular interpolation (which can be improved further with a cubic option). To do this, just use the Curve.interpolate_baked() method together with Curve2D.get_baked_length(). The first call to either of them will bake the curve internally.
Traversal at constant speed, then, can be done with the following pseudo-code:
And the output will, then, move at constant speed: