An orthogonal projection is a projection of a figure by parallel rays perpendicular to the target plane or subspace. For a planar figure whose projection is nondegenerate, tangencies are preserved, parallel lines project to parallel lines, and ratios of lengths of parallel segments and of areas are preserved.
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 a suitable affine transformation, a triangle and its Steiner inellipse can be transformed into a circle inscribed in an equilateral triangle.
Spheroids project to ellipses (with circles as a special case).
For an orthogonal projection 
onto a subspace 
, any vector 
in an inner product space
can be written 
,
so
 |
and, with respect to a real orthonormal basis, the projection matrix is a symmetric matrix iff the vector space projection is orthogonal. With respect to a complex orthonormal basis, the corresponding condition is that the projection matrix is a Hermitian matrix.
