Orthographic Projection — Definition, Formula & Examples
Orthographic projection is a way of mapping a point or shape onto a line or plane by dropping a perpendicular from each point to that line or plane. The result is a flat "shadow" that preserves parallel lines and true dimensions along axes parallel to the projection plane.
An orthographic projection is a parallel projection in which the lines of projection are perpendicular (orthogonal) to the projection plane. For a point projected onto the -plane, the image is the point , obtained by setting the component normal to the plane equal to zero.
Key Formula
Where:
- = Original point in 3D space
- = Projected image of P on the xy-plane
- = Component perpendicular to the projection plane, set to 0
How It Works
Pick the plane you want to project onto — commonly one of the coordinate planes such as , , or . For every point in your figure, draw a line perpendicular to that plane. Where the perpendicular line meets the plane is the projected image of the point. Because every projection line is parallel to every other, the projection preserves parallelism and ratios of lengths along directions parallel to the plane. Distances along the direction perpendicular to the plane are lost entirely.
Worked Example
Problem: Find the orthographic projection of the point A = (3, 5, 8) onto the xy-plane.
Identify the perpendicular direction: The xy-plane has the equation z = 0, so the perpendicular direction is along the z-axis.
Drop the z-coordinate: Set the z-component of A to zero to get the projected point.
Answer: The orthographic projection of A onto the xy-plane is A' = (3, 5, 0).
Why It Matters
Orthographic projection is the basis for engineering and architectural drawings, where top, front, and side views each show true measurements along two axes. In linear algebra and computer graphics courses, it introduces the idea of projecting vectors onto subspaces, a skill used throughout higher mathematics and 3D modeling.
Common Mistakes
Mistake: Confusing orthographic projection with perspective projection, which makes distant objects appear smaller.
Correction: In orthographic projection, all projection lines are parallel and perpendicular to the plane — there is no foreshortening based on distance from a viewpoint.