GeometricTransformation

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`GeometricTransformation`

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GeometricTransformation[g,tfun]

represents the result of applying the transformation function tfun to the geometric objects corresponding to the primitives g.

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GeometricTransformation[g,m]

transforms geometric objects in g by effectively replacing every point r by m.r.

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GeometricTransformation[g,{m,v}]

effectively replaces every point r by m.r+v.

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GeometricTransformation[g,{t₁,t₂,…}]

represents multiple copies of g transformed by a collection of transformations.

Details and Options

GeometricTransformation[g,…] remains unchanged under evaluation, but affects how g is rendered.
GeometricTransformation works on lists of graphics primitives and directives in 2D and 3D.
GeometricTransformation[g,{m,v}] effectively applies an affine transform to g.
GeometricTransformation[g,{{m_xx,m_yx},{m_xy,m_yy}}] transforms the unit vectors and to {m_xx,m_xy} and {m_yx,m_yy}, respectively.
For different spec, GeometricTransformation[g,{m,spec}] leaves fixed the following special points on the bounding box of g:

	Center	center
	Left	midpoint of the left side
	Right	midpoint of the right side
	Top	midpoint of the top
	Bottom	midpoint of the bottom
	Front	midpoint of the front
	Back	midpoint of the back
	{Left,Top}, etc.	corners

For objects specified with scaled coordinates Scaled[{x,y}], GeometricTransformation effectively applies its transformation to the corresponding ordinary coordinates.
Normal[expr] if possible replaces all GeometricTransformation[g_i,…] constructs by versions of the g_i in which the coordinates have explicitly been transformed.
The following option can be given:
ContentSelectable Automatic whether to allow contents to be selected
For matrices m₁ and m₂, GeometricTransformation[GeometricTransformation[g,m₁],m₂] is equivalent to GeometricTransformation[g,m₂.m₁].

Examples

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Basic Examples (3)Summary of the most common use cases

Transform a 2D object:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-bqzeyc

Out[1]=1

Transform a 3D object:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-cc647u

Out[1]=1

Multiple transforms can be applied to the same object:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-e7l85t

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dc16foc4fjq-by83yw

Out[2]=2

Scope (5)Survey of the scope of standard use cases

Transformation applied to a 2D shape:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-cp0s4d

In[2]:=2

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https://wolfram.com/xid/0dc16foc4fjq-bx2faq

Out[2]=2

Transformation applied to a 3D shape:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-guhj3r

In[2]:=2

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https://wolfram.com/xid/0dc16foc4fjq-cikv87

Out[2]=2

Objects with scaled coordinates:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-ccfue9

Out[1]=1

Keep the rightmost point of the circle fixed:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-fb0o2b

Out[1]=1

Create nested transformations:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-dbo3du

Out[1]=1

Properties & Relations (2)Properties of the function, and connections to other functions

Using {m,v} as the second argument is the same as using AffineTransform[{m,v}]:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-0rx4j

Out[1]=1

When possible, Normal will perform the transformations explicitly:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-lgap6l

Out[1]=1

Neat Examples (1)Surprising or curious use cases

Rotating and moving a cuboid along a space curve:

In[1]:=1

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https://wolfram.com/xid/0dc16foc4fjq-d0ce8c

Out[1]=1

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