Orthogonal Projection

A projection of a figure by parallel rays. In such a projection, tangencies are preserved. Parallel lines project to parallel lines. The ratio of lengths of parallel segments is preserved, as is the ratio of areas.

Any triangle can be positioned such that its shadow under an orthogonal projection is equilateral. Also, the triangle medians of a triangle project to the triangle medians of the image triangle. Ellipses project to ellipses, and any ellipse can be projected to form a circle. The center of an ellipse projects to the center of the image ellipse. The triangle centroid of a triangle projects to the triangle centroid of its image. Under an orthogonal transformation, the Steiner inellipse can be transformed into a circle inscribed in an equilateral triangle.

Spheroids project to ellipses (or circles in the degenerate case).

In an orthogonal projection, any vector v can be written v=v_W+v_(W^_|_), so


and the projection matrix is a symmetric matrix iff the vector space projection is orthogonal.

See also

Projection, Projection Matrix

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Orthogonal Projection." From MathWorld--A Wolfram Web Resource.

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