orx-shapes/org.openrndr.extra.shapes.ph/PHQuintic

PHQuintic

A planar Pythagorean-hodograph (PH) quintic: a quintic Bézier curve whose hodograph, its derivative, is the square of a complex quadratic, r'(t) = w(t)², with w(t) = w0·(1-t)² + w1·2t(1-t) + w2·t². Its parametric speed |r'(t)| = |w(t)|² is therefore a polynomial, so unlike for ordinary Béziers the arc length is a polynomial too, the unit tangent and normal are rational, and so are its offset curves. See Farouki, "Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable" (Springer, 2008).

Positive offsets lie on the side of the normal (-y, x) of the direction of travel, which is to the right in a y-down frame, as for org.openrndr.extra.shapes.arcer.arcer.

Properties

controlPoints

val controlPoints: List<Vector2>

The six control points of the quintic Bézier.

end

val end: Vector2

The end point.

length

val length: Double

The exact arc length.

start

val start: Vector2

The start point.

Functions

arcLength

fun arcLength(t: Double): Double

The exact arc length from the start to parameter t.

contour

fun contour(offset: Double = 0.0, tolerance: Double = 0.01, polarity: YPolarity = YPolarity.CW_NEGATIVE_Y): ShapeContour

The curve, or its offset at offset, as a contour of cubic Béziers within tolerance, see segments.

curvature

fun curvature(t: Double): Double

The signed curvature 2·Im(conj(w)·w') / |w|⁴, positive where the curve turns towards the normal.

derivative

fun derivative(t: Double): Vector2

The derivative r'(t) = w(t)².

normal

fun normal(t: Double): Vector2

The unit normal (-y, x) of the tangent.

offset

fun offset(t: Double, distance: Double): Vector2

The exact point of the offset curve at distance distance along the normal.

offsetDerivative

fun offsetDerivative(t: Double, distance: Double): Vector2

The derivative of the offset curve, r'(t)·(1 - distance·κ(t)).

parameterAtLength

fun parameterAtLength(length: Double): Double

The parameter at which the arc length from the start equals length, by Newton's method on the polynomial.

position

fun position(t: Double): Vector2

The position at parameter t in 0, 1.

segments

fun segments(offset: Double = 0.0, tolerance: Double = 0.01): List<Segment2D>

Cubic Bézier segments that approximate the curve, or its offset at offset, within tolerance. Every piece matches the position and derivative exactly at both ends, pieces are halved until their midpoint deviates less than tolerance.

speed

fun speed(t: Double): Double

The parametric speed |r'(t)| = |w(t)|², a polynomial.

tangent

fun tangent(t: Double): Vector2

The unit tangent w²/|w|², rational in t.