Typescript Math Toolkit Quadratic Bezier
This is a beta release of the Typescript Math Toolkit quadratic bezier class, complete with arc-length parameterization. When I first began programming Bezier curves in the 1980s, I shared the same general impression concerning the natural parameter, t, with most programmers. It's easy to presume that the point on the curve at t = 0.5, for example, is always at exactly one-half of the total length of the curve. This is, in fact, only true under very limited circumstances. In general, let s represent normalized arc length in [0,1]. So, s = 0 is the initial point of the curve and s = 1 is the terminal point of the curve, i.e. at 100% of the total arc length of the curve. Let p = B(t) and q = B(s) represent points on the curve at some value of the natural parameter, t and normalized arc-length, s, where both t and s are in the open interval (0,1). In general, p != q. The points may be close or notably distinct, depending on the location of the curve's control points or geometric constraints.
Arc-length parameterization is required for applications that must identify points at a specific fraction of the curve's total length. This parameterization is performed numerically as there are no simple, closed-form formulas that produce a naturally arc-length parameterized curve. Every Bezier code I've written since the 1990's has such parameterization and the current Typescript implementation is no different.
The current quad Bezier library is illustrated via an interactive Angular 6 demo that uses PIXI 4 for dynamic drawing. This demo differs from others in that three interpolation points are specified instead of the typical three control points. Three-point quad Bezier interpolation is applied to compute geometric constraints that cause the quad Bezier to pass through all three input points.
Now, there are an infinity of quadratic curves that pass through three points that are distinct and not collinear. In order to specify a unique curve, a parameter value (t) must be chosen so that the Bezier curve passes through the middle interpolation point at that value of the natural parameter. The quad Bezier class computes this parameter as the fraction of chord length from the first to second points divided by the sum of the chord length from first to second and second to third points. The demo draws the Bezier curve and the geometric constraints that cause the curve to interpolate all three points.
Points on the curve at equal increments of t and s may also be illustrated. Drag one of the control points to see how the relationship between the two varies as a function of the geometric constraints.
Author: Jim Armstrong - The Algorithmist
theAlgorithmist [at] gmail [dot] com
Angular CLI: 6.2.3
Running the demo
The drawing area is represented in light blue. Click anywhere in that area to define three interpolation points, which are visually represented by red circles. After the third point is entered, the display is updated to show the control points and the quadratic Bezier curve. The total arc length of the curve is displayed as well. Click on the Equal t and/or Equal s checkboxes to visualize points on the curve at t = 0.2, 0.4, 0.6, 0.8 as or s = 0.2, 0.4, 0.6, 0.8. Drag one of the control points to see how the spread between natural and normalized arc length parameterization changes as a function of geometric constraints.
The quad Bezier public API is very robust and is summarized below. This summary does not include other methods available in the PlanarCurve superclass.
fromObject(control: IControlPoints): void public getX(t: number): number public getY(t: number): number public getXPrime(t: number): number public getYPrime(t: number): number public interpolate(points: Array<IPoint>): number public getTAtS(s: number): number public getTAtX(x: number): Array<number> public getYAtX(x: number): Array<number> public getXAtY(y: number): Array<number> public sAtT(t: number): number
Since numerical approximation is required for general arc-length parameterization as well as some of the internal computations, the code distribution also includes classes for numerical integration and root finding.
ng serve for a dev server. Navigate to
http://localhost:4200/. The app will automatically reload if you change any of the source files.
ng build to build the project. The build artifacts will be stored in the
dist/ directory. Use the
--prod flag for a production build.
To get more help on the Angular CLI use
ng help or go check out the Angular CLI README.
Free Software? Yeah, Homey plays that