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swampwiz
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I was reading about the homothetic transformation, and it seems that the perspective transform is a type of this.
It does say it. They define a homothetic transformation as ##M \mapsto \lambda \cdot \stackrel{\longrightarrow}{SM}##. That's exactly what a perspective does to a point ##M## as seen from ##S##.swampwiz said:It doesn't seem to say that a perspective transform is a type of homothetic, although it sure looks like it.
That's what I thought.fresh_42 said:It does say it. They define a homothetic transformation as ##M \mapsto \lambda \cdot \stackrel{\longrightarrow}{SM}##. That's exactly what a perspective does to a point ##M## as seen from ##S##.
They are affine transformations. The difference to linear transformations is only whether ##S=0## or ##S\neq 0##.swampwiz said:That's what I thought.
Now, what about affine transformations? Is the set of all Homothetic transformations also affine transformations? or vice-versa? It seems that since lines are preserved in a homothetic transformation, it is also an affine transformation.
In reality there is something like resolution. Although it might not exist theoretically, there is a real margin below which we have indistinguishability.swampwiz said:I will need to look at it along the line in the exact same direction of the line, so there should never be a vanishing point, ...
No, it's affine linear, i.e. the origin is the center of projection and not the origin of the coordinate system.... and essentially that a proper perspective projection is not linear.
Perspective in mathematics is a way of drawing or representing three-dimensional objects on a two-dimensional surface, such as a piece of paper. It involves creating the illusion of depth and distance by using vanishing points and converging lines.
A homothetic transformation is a type of transformation in geometry where an object is scaled or enlarged by a constant factor in all directions. This means that all the distances within the object are multiplied by the same number, resulting in a similar but larger or smaller version of the original object.
Perspective can be thought of as a type of homothetic transformation, as it involves scaling an object in a way that maintains its proportions and creates the illusion of depth and distance. In perspective drawing, the size of an object decreases as it moves away from the viewer, similar to how a homothetic transformation would scale an object.
No, not all perspective drawings can be considered homothetic transformations. While perspective involves scaling an object, it also involves other techniques such as foreshortening and vanishing points. Homothetic transformations, on the other hand, only involve scaling an object by a constant factor.
Homothetic transformations and perspective have many real-life applications, including in architecture, engineering, and art. In architecture and engineering, they are used to create accurate and realistic representations of buildings and structures. In art, they are used to create the illusion of depth and distance, as well as to create aesthetically pleasing compositions.