Hidden line

In mathematics, a hidden line is a geometric edge line[1] 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,[2] or thinner lines than the visible lines.

Hidden lines add geometric information about the unseen sides of an object. They are used to help a person visualize drawings of geometric objects in three-dimensional space.[3]

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.

Geometry 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.[4] Less lines makes an object easier to draw.

Cube projection graph, in a B3 Coxeter plane. The cube rotated. The cube without hidden lines.
Tesseract projection graph, in a B4 Coxeter plane. Tesseract with visible lines highlighted and hidden lines as dash lines. The tesseract without hidden lines.
5 dimensional hypercube graph, in a B5 Coxeter plane. The hypercube rotated with visible lines and vertex highlighted. 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.

Hidden lines in technical drawings

Hidden lines represent edges of a physical object that are not visible from a specific viewpoint. Visible lines are viewable edges from the specified viewpoint. Visible lines are drawn as solid lines and hidden lines are drawn as lines of short dashes evenly spaced.[5] Hidden lines are particularly useful when viewing an object from an angle where the visible only lines do not have much information. In a computer drawing application the option may be referred to a wire frame with hidden edges.[6] The hidden edges would be dashed lines.

A wire-frame image using hidden-line removal

Removing hidden lines is 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

References

  1. ^ Weisstein, Eric W. "Polytope Edge". From Wolfram MathWorld.
  2. ^ TeX software(TeX), Draw cube with dashed hidden lines. From TeX StackExchange.
  3. ^ "...needed to make the drawing clear", Hidden Lines and Centerlines, Modern Graphics Communication, 5th Edition
  4. ^ "Hidden lines are omitted from pictorial drawings unless they are needed...", Hidden Lines and Centerlines, Modern Graphics Communication, 5th Edition
  5. ^ Engineering Design Graphics > Line Conventions and Lettering> Hidden Lines, Robert Hewitt, Associate Professor at the University of Memphis
  6. ^ Line Types in Technical Drawings YouTube video