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Hidden line

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In mathematics, a hidden line is a geometric edge lineWeisstein, Eric W. "[http://mathworld.wolfram.com/PolytopeEdge.html Polytope Edge]". From Wolfram MathWorld. that is not visible from an observer's view of a shape or object.

A common practice is to draw the visible edges as solid lines and the hidden lines as dotted lines, dashed lines,{{Stylized LaTeX|4}} software(TeX), [https://tex.stackexchange.com/questions/198977/how-to-draw-cubes-in-pstricks-with-dashed-hidden-lines Draw cube with dashed hidden lines]. From TeX StackExchange. or thinner lines than the visible lines.

File:Hidden Line Cubes.jpg with hidden lines: dotted, dashed, and thin lines. Cubes with no hidden lines, and a cube with dotted lines to give 3 dimensional reference]]

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."...needed to make the drawing clear", [https://www.peachpit.com/articles/article.aspx?p=2873372&seqNum=15 Hidden Lines and Centerlines], Modern Graphics Communication, 5th Edition

File:Wire frame.svg 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."Hidden lines are omitted from pictorial drawings unless they are needed...", [https://www.peachpit.com/articles/article.aspx?p=2873372&seqNum=15 Hidden Lines and Centerlines], Modern Graphics Communication, 5th Edition Less lines makes an object easier to draw.

{{multiple image

| footer = Cube projection graph, in a B3 Coxeter plane. The cube rotated. The cube without hidden lines.

| image1 = 3-cube column graph.svg

| image2 = Cube With Hidden Lines.jpg

| image3 = Cube Without Hidden Lines.jpg

| total_width = 240px

}}

{{multiple image

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

| image1 = 4-cube_t0.svg

| image2 = Tesseract_With_Hidden_Dash_Lines.jpg

| image3 = Tesseract_Without_Hidden_Lines.jpg

| total_width = 240px

}}

{{multiple image

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

| image1 = 5-cube_graph.svg

| image2 = 5 Dimensional Hypercube With Hidden Lines.jpg

| image3 = 5 Dimensional Hypercube Without Hidden Lines.jpg

| total_width = 240px

}}

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 a specified viewpoint. In a technical drawing hidden lines are either included or not included, depending on the viewer’s requirement.

"Hidden lines are used in engineering drawings to represent features that cannot be seen in a particular view but are necessary to fully define the part or assembly. Hidden lines are represented by a series of short dashes, evenly spaced, with the first dash in contact with the visible line from which it starts and the last dash in contact with the visible line at which it ends."[https://engineeringtechnology.org/engineering-graphics/line-conventions-and-lettering/hidden-lines/ Engineering Design Graphics, Line Conventions, and Hidden Lines], Robert Hewitt, Associate Professor at the University of Memphis

In a computer drawing application, the hidden line option may be referred to as a wire frame with hidden edges.[https://www.youtube.com/watch?v=_t2NjPpsegE&t=64s Line Types in Technical Drawings] YouTube video The wireframe hidden edges would be dashed lines.

File:Obj lineremoval.png applied.]]

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. "The general problem of efficiently and meaningfully displaying three-dimensional objects in two dimensions is central to computer graphics."[https://arxiv.org/abs/0809.4093 Perspective Drawing of Surfaces with Hidden Line Removal(in Spanish)] [https://github.com/tigerfarm/tfpscience/blob/main/hiddenline/hiddenLineTrans1a.pdf Perspective Drawing of Surfaces with Hidden Line Removal(PDF English translation)]

File:Point of View Mapping of a 3D Surface to 2D.jpg

See also

  • Hidden-surface determination
  • Technical drawing
  • [https://arxiv.org/abs/0809.4093 Perspective Drawing of Surfaces with Hidden Line Removal] The [https://arxiv.org/ arXiv] article is written in Spanish. [https://github.com/tigerfarm/tfpscience/blob/main/hiddenline/hiddenLineTrans1a.pdf PDF English translation].

References

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External links

  • [https://www.linkedin.com/advice/0/what-advantages-disadvantages-using-dashed-dotted What are the advantages and disadvantages of using dashed or dotted hidden lines?]

Category:Elementary geometry

Category:Multi-dimensional geometry

Category:Graphical projections

Category:Technical drawing