Affine linear transformation of 3D objects

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plot::Transform3d(<b3d>, A3d, obj1, <obj2,...>, <a = amin .. amax>, options)


plot::Transform3d(b, A, objects) with a vector b and a matrix A applies the affine linear transformation xAx+b to 3D objects.

The transformation matrix A may be specified by a list of lists, with the sublists representing the rows:

[[A1, 1, A1, 2, …], [A2, 1, A2, 2, …], …]

A plain list [A1, 1, A1, 2, …, A3, 2, A3, 3] represents the matrix row by row.

Transform objects can transform several graphical objects simultaneously. Plotting the transform object renders all graphical objects inside.

Transformed objects have a tendency to overestimate their ViewingBox. In such cases, specify a suitable ViewingBox explicitly.

Transformation objects can be used inside transformation objects. If they are animated, the animations run simultaneously.

Animated transform objects are rather “cheap” concerning computing and storing costs. For more complex graphical objects, it is more efficient to use an animated transform object than to redefine the object for each frame.

The function op allows to extract the graphical objects inside a transformation object.


AttributePurposeDefault Value
AffectViewingBoxinfluence of objects on the ViewingBox of a sceneTRUE
Framesthe number of frames in an animation50
Matrix3dtransformation matrices[1, 0, 0, 0, 1, 0, 0, 0, 1]
Namethe name of a plot object (for browser and legend) 
ParameterEndend value of the animation parameter 
ParameterNamename of the animation parameter 
ParameterBegininitial value of the animation parameter 
ParameterRangerange of the animation parameter 
Shiftshift vector[0, 0, 0]
ShiftXshift vector0
ShiftYshift vector0
ShiftZshift vector0
TimeEndend time of the animation10.0
TimeBeginstart time of the animation0.0
TimeRangethe real time span of an animation0.0 .. 10.0


Example 1

For some applications, it is very popular to plot a function in 3D together with a projection of its contour lines onto the lower or upper bounding plane. MuPAD® has no direct option for this, but with plot::Transform3d, it is possible to achieve the same effect. Suppose that you have the function under consideration in a plot::Function3d object:

f := plot::Function3d(sin(x*y)+cos(x^2-y),
                      x=-3..3, y=-3..3, Submesh=[1,1]):

To plot contour lines at all, use the attribute ZContours. To avoid changing f, create a modified copy using plot::modify:

plot(plot::modify(f, ZContours = [Automatic, 10]))

To only get contour lines, change a few more parameters: switch off the surface and the parameter lines. Then, add height coloring to the lines and use plot::Transform3d to project them onto the plane z = - 2.5. Finally, plot these lines together with the original function:

     plot::Transform3d([0, 0, -2.5], // shift vector
                       [1, 0, 0,  // transformation matrix
                        0, 1, 0,
                        0, 0, 0],
                    Filled = FALSE,
                    XLinesVisible = FALSE, YLinesVisible = FALSE,
                    ZContours = [Automatic, 10],
                    LineColorFunction =  // height coloring
                ((x, y, z) -> [(z+2)/4, 0, (2-z)/4]))))



The 3D shift vector: a list with 3 entries. Also vectors generated by matrix or arrays are accepted. The entries must be numerical values or arithmetical expressions of the animation paramater a.

b3d is equivalent to the attribute Shift.


The 3D transformation matrix: a 3×3 matrix, a 3×3 array, a list of 3 lists, or a plain list with 9 entries. The entries must be numerical values or arithmetical expressions of the animation paramater a.

A3d is equivalent to the attribute Matrix3d.

obj1, obj2, …

Plot objects


Animation parameter, specified as a = amin..amax, where amin is the initial parameter value, and amax is the final parameter value.