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which you might recognise as an application of the Pythagorean Theorem. Maps that preserve the distance between all pairs of points are called isometries, and include reflections, rotations, translations, and compositions thereof. In matrix notation any of these have the form |
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which you might recognise as an application of the Pythagorean Theorem. This turns Rn) into a metric space. Maps that preserve the distance between all pairs of points are called isometries, and include reflections, rotations, translations, and compositions thereof. In matrix notation any of these have the form |
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where A is a [special orthogonal matrix]? and b is a column vector. Isometries are taken as the congruences of Euclidean geometry - that is, we only consider properties preserved by them. That way we do not have to worry about the precise origin or axes, but still consider distances, angles, and so forth. |
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where A is an [orthogonal matrix]? and b is a column vector. Isometries are taken as the congruences of Euclidean geometry - that is, we only consider properties preserved by them. That way we do not have to worry about the precise origin or axes, but still consider distances, angles, and so forth. |
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