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Affine linear transformations with
a vector *b* and
a matrix *A* can
be applied to graphical objects via transformation objects. There
are special transformations such as translations, scaling, and rotations
as well as general affine linear transformations:

`plot::Translate2d([b1, b2], Primitive1, Primitive2, ...)`

applies the translation by the vector*b*=`[b1, b2]`

to all points of 2D primitives.`plot::Translate3d([b1, b2, b3], Primitive1, ...)`

applies the translation by the vector*b*=`[b1, b2, b3]`

to all points of 3D primitives.`plot::Reflect2d([x1, y1], [x2, y2], Primitive1, ...)`

reflects all 2D primitives about the line through the points`[x1, y1]`

and`[x2, y2]`

.`plot::Reflect3d([x, y, z], [nx, ny, nz], Primitive1, ...)`

reflects all 3D primitives about the plane through the point`[x, y, z]`

with the normal`[nx, ny, nz]`

.`plot::Rotate2d(angle, [c1, c2], Primitive1, ...)`

rotates all points of 2D primitives counter clockwise by the given angle about the pivot point`[c1, c2]`

.`plot::Rotate3d(angle, [c1, c2, c3], [d1, d2, d3], Primitive1, ...)`

rotates all points of 3D primitives by the given angle around the rotation axis specified by the pivot point`[c1, c2, c3]`

and the direction`[d1, d2, d3]`

.`plot::Scale2d([s1, s2], Primitive1, ...)`

applies the diagonal scaling matrix`diag`

(`s1`

,`s2`

) to all points of 2D primitives.`plot::Scale3d([s1, s2, s3], Primitive1, ...)`

applies the diagonal scaling matrix`diag`

(`s1`

,`s2`

,`s3`

) to all points of 3D primitives.`plot::Transform2d([b1, b2], A, Primitive1, ...)`

applies the general affine linear transformation with a 2×2 matrix`A`

and a vector*b*=`[b1, b2]`

to all points of 2D primitives.`plot::Transform3d([b1, b2, b3], A, Primitive1, ...)`

applies the general affine linear transformation with a 3×3 matrix`A`

and a vector*b*=`[b1, b2, b3]`

to all points of 3D primitives.

The ellipses `plot::Ellipse2d`

provided
by the `plot`

library have axes parallel to the coordinate
axes. We use a rotation to create an ellipse with a different orientation:

center := [1, 2]: ellipse := plot::Ellipse2d(2, 1, center): plot(plot::Rotate2d(PI/4, center, ellipse))

Transform objects can be animated. We build a group consisting of the ellipse and its symmetry axes. An animated rotation is applied to the group:

g := plot::Group2d( ellipse, plot::Line2d(center, [center[1] + 2, center[2]]), plot::Line2d(center, [center[1] - 2, center[2]]), plot::Line2d(center, [center[1], center[2] + 1]), plot::Line2d(center, [center[1], center[2] - 1]) ): plot(plot::Rotate2d(a, center, a = 0..2*PI, g)):

Objects inside an animated transformation can be animated, too.
The animations run independently and may be synchronized via suitable
values of the `TimeRange`

as described in section Advanced Animations: The Synchronization Model.

We generate a sphere *s* of
radius *r* with
center *c* = (*c*_{x}, *c*_{y}, *c*_{z}).
We wish to visualize the tangent plane at various points of the surface.
We start with the tangent plane of the north pole and rotate it around
the *y* axes
(i.e., along the line with zero longitude) by the polar angle *θ* for
the first 3 seconds. Then it is rotated around the *z*-axis
(i.e., along the line with constant latitude) by the azimuth angle *ϕ*.
We end up with the tangent plane at the point *x* = *c*_{x} + *cos*(*ϕ*) *sin*(*θ*), *y* = *c*_{y} + *sin*(*ϕ*) *sin*(*θ*), *z* = *c*_{z} + *cos*(*θ*).
The two rotations are realized as a nested animation: By specifying
disjoint time ranges, the second rotation (around the *z*-axis)
starts when the first rotation (around the *y*-axis)
is finished:

r := 1: // the radius of the sphere R := 1.01: // increase the radius a little bit c := [0, 0, 0]: // the center of the sphere thet := PI/3: // spherical coordinates of phi := PI/4: // the final point p // the final point: p := plot::Point3d(c[1] + R*cos(phi)*sin(thet), c[2] + R*sin(phi)*sin(thet), c[3] + R*cos(thet), PointSize = 2*unit::mm, Color = RGB::Black): // the sphere: s := plot::Sphere(r, c, Color = RGB::Green): // the meridian at thet = 0 c1 := plot::Curve3d([c[1] + R*sin(t), c[2], c[3] + R*cos(t)], t = 0..thet, Color = RGB::Black): // the meridian at thet = 0 c2 := plot::Curve3d([c[1] + R*cos(t)*sin(thet), c[2] + R*sin(t)*sin(thet), c[3] + R*cos(thet)], t = 0..phi, Color = RGB::Black): // form a group consisting of the tangent plane and its normal: g := plot::Group3d( plot::Surface([c[1] + u, c[2] + v, c[3] + R], u = -1..1, v = -1..1, Mesh = [2, 2], Color = RGB::Red.[0.3]), plot::Arrow3d([c[1], c[2], c[3] + R], [c[1], c[2], c[3] + R + 0.7]) ): // rotate the group for 3 seconds along the meridian: g := plot::Rotate3d(a, c, [0, 1, 0], a = 0..thet, g, TimeRange = 0..3): // rotate the group for further 3 seconds along the azimuth: g := plot::Rotate3d(a, c, [0, 0, 1], a = 0..phi, g, TimeRange = 3..6): plot(s, g, c1, c2, p, CameraDirection = [2, 3, 4]):

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