Orthogonal projection in space and its use in engineering
If the direction of projection is perpendicular to the plane of projection, the parallel projection is called orthogonal; all properties of parallel projection (preservation of parallelism and ratios) hold. An extra property of orthogonal projection: the area S′ of the projection of a polygon equals the polygon’s area S times the cosine of the angle φ between its plane and the plane of projection: S′ = S · cos φ. For a triangle with one side in the plane of projection, the altitude to that side shrinks cos φ times in projection by the theorem of three perpendiculars while the base is unchanged; a general polygon is cut into triangles. Likewise a segment AB making angle α with the plane has projection length AB · cos α. In technical drawing a part is orthogonally projected onto two or three mutually perpendicular planes (front, side and top views).
Hold a cardboard rectangle at different tilts under a desk lamp (the lamp may be hot, so keep an adult nearby) or in sunlight, measure the shadow dimensions and check that one dimension is multiplied by the cosine of the tilt.