Affine Transformations

Transform rotate (const real x, [const real y = 0, [const real z = 0]])	Function
Transform rotate (const Point& p0, const Point& p1, [const real angle = 180])	Function
Transform rotate (const Path& p, [const real angle = 180])	Function

Each of these functions calls the corresponding version of Transform::rotate(), and returns its return value, namely, a Transform representing the rotation only. In the first version, taking three real arguments, the Point is rotated x degrees around the x-axis, y degrees around the y-axis, and z degrees around the z-axis in that order. Point p0(1, 0, 2); p0.rotate(90); p0.show("p0:") -| p0: (1, 2, 0) Point p1(-1, 1, 1); p1.rotate(-90, 90, 90); p1.show("pt1:"); -| p1: (1, -1, -1) Fig. 81. Please note that rotations are not commutative operations. Nor are they commutative with other transformations. So, if you want to rotate a Point about the x, y and z-axes in that order, you can do so with a single invocation of rotate(), as in the previous example. However, if you want to rotate a Point first about the y-axis and then about the x-axis, you must invoke rotate() twice. Point pt0(1, 1, 1); pt0.rotate(0, 45); pt0.rotate(45); pt0.show("pt0:"); -| pt0: (0, 1.70711, 0.292893) In the version taking two Point arguments p0 and p1, and a real argument angle, the Point is rotated angle degrees around the axis determined by p0 and p1, 180 degrees by default. Point P(2, 0, 0); Point A; Point B(2, 2, 2); P.rotate(A, B, 180); Fig. 82.

Transform scale (real x, [real y = 1, [real z = 1]])

Function

Calls transform.scale(x, y, z) and returns its return value, namely, a Transform representing the scaling operation only. Scaling causes the x-coordinate of the Point to be multiplied by x, the y-coordinate of the Point to be multiplied by y, and the z-coordinate of the Point to be multiplied by z. Point p0(1, 0, 3); p0.scale(4); p0.show("p0:"); -| p0: (4, 0, 3) Point p1(-2, -1, -2); p1.scale(-2, -3, -4); p1.show("p1:"); -| p1: (4, 3, 8) Fig. 83.

Transform shear (real xy, [real xz = 0, [real yx = 0, [real yz = 0, [real zx = 0, [real zy = 0]]]]])

Function

Calls transform.shear() with the same arguments and returns its return value, namely, a Transform representing the shearing operation only. Shearing modifies each coordinate of a Point proportionately to the values of the other two coordinates. Let x_0, y_0, and z_0 stand for the coordinates of a Point P before P.shear(\alpha, \beta, \gamma, \delta, \epsilon, \zeta ), and x_1, y_1, and z_1 for its coordinates afterwards. x_1 == x_0 + \alpha y + \beta z y_1 == y_0 + \gamma x + \delta z z_1 == z_0 + \epsilon x + \zeta y [next figure] demonstrates the effect of shearing the four Points of a 3 * 3 Rectangle (i.e., a square) r in the x-y plane using only an xy argument, making it non-rectangular. Point P0; Point P1(3); Point P2(3, 3); Point P3(0, 3); Rectangle r(p0, p1, p2, p3); r.draw(); r.shear(1.5); r.draw(black, "evenly"); Fig. 84.

Transform shift (real x, [real y = 0, [real z = 0]])	Function
Transform shift (const Point& p)	Function

Each of these functions calls the corresponding version of Transform::shift() on transform, and returns its return value, namely, a Transform representing the shifting operation only. The Point is shifted x units in the direction of the positive x-axis, y units in the direction of the positive y-axis, and z units in the direction of the positive z-axis. p0(1, 2, 3); p0.shift(2, 3, 5); p0.show("p0:"); -| p0: (3, 5, 8)

Transform shift_times (real x, [real y = 1, [real z = 1]])	Function
Transform shift_times (const Point& p)	Function

Each of these functions calls the corresponding version of Transform::shift_times() on transform and returns its return value, namely the new value of transform. shift_times() makes it possible to increase the magnitude of a shift applied to a Point, while maintaining its direction. Please note that shift_times() will only have an effect if it's called after a call to shift() and before transform is reset. This is performed by reset_transform(), which is called in apply_transform(), and can also be called directly. See Transform Reference; Resetting, and Point Reference; Applying Transformations. Point P; P.drawdot(); P.shift(1, 1, 1); P.drawdot(); P.shift_times(2, 2, 2); P.drawdot(); P.shift_times(2, 2, 2); P.drawdot(); P.shift_times(2, 2, 2); P.drawdot(); Fig. 85.