Hidden line

In mathematics, a hidden line is a geometric edge line that is not visible from an observer's view of a shape or object.

Cubes with hidden lines: dotted, dashed, and thin lines. Cubes with no hidden lines, and a cube with dotted lines to give 3 dimensional reference

A common practice is to draw the visible edges as solid lines and the hidden lines as dotted lines, dashed lines, or thinner lines than the visible lines.

Hidden lines can help visualize geometric objects in three-dimensional space. Hidden lines add geometric information about the unseen sides of an object.

Sample rendering of a wire-frame cube, icosahedron, and sphere

A three-dimensional object drawn with solid visible and hidden lines is a wire-frame model of the object.

Objects without hidden lines

When a student is first introduced to a geometric cube, a cube without hidden lines looks like a closed box which they are familiar. Drawing an object without hidden lines, matches an observer's view of the object. Less lines makes an object easier to draw.

Cube projection graph, in a B₃ Coxeter plane, the cube rotated, and the cube without hidden lines.

Tesseract projection graph, in a B₄ Coxeter plane, tesseract with visible lines highlighted and hidden lines as dash lines, and the tesseract without hidden lines.

5 dimensional hypercube graph, in a B₅ Coxeter plane, the hypercube rotated with visible lines and vertex highlighted, and the hypercube without hidden lines.

To draw higher dimensional cubes, hypercubes, without hidden lines, make the faces opaque. Then, the hidden lines are no longer visible, they are removed from the observer's view. This works with a cube in three-dimensional space, a Tesseract in four-dimensional space, a hypercube in five-dimensional space, and will work with higher dimensions as well.

A wire-frame image using hidden-line removal

Removing hidden lines is also important in computer design and graphics applications. There are algorithmic solutions to remove hidden lines or partially hidden lines during an object's rendering.

See also

• Hidden-surface determination
• Technical drawing