In mathematics, a hidden line is a geometric edge line that is not visible from an

observer An observer is one who engages in observation or in watching an experiment. Observer may also refer to: Fiction * ''Observer'' (novel), a 2023 science fiction novel by Robert Lanza and Nancy Kress * ''Observer'' (video game), a cyberpunk horr ...

'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, or thinner lines than the visible lines. Hidden Line Cubes

Hidden lines add geometric information about the unseen sides of an object. They are used to help a person visualize drawings of

geometric Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

objects in

three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...

A three-dimensional object drawn with solid visible and hidden lines is a

wire-frame model In 3D computer graphics, a wire-frame model (also spelled wireframe model) is a visual representation of a three-dimensional (3D) physical object. It is based on a polygon mesh or a volumetric mesh, created by specifying each Edge (geometry ...

of the object.

Geometry objects without hidden lines

When a student is first introduced to a geometric

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

, 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. To draw higher dimensional cubes,

hypercubes In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square (two-dimensional, ) and a cube (Three-dimensional, ); the special case for Four-dimensional space, is known as a ''tesseract''. It is ...

, without hidden lines, make the

faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...

opaque. Then, the hidden lines are no longer visible, they are removed from the observer's view. This works with a

, a

Tesseract In geometry, a tesseract or 4-cube is a four-dimensional hypercube, analogous to a two-dimensional square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and the surface of the cube consists of six ...

four-dimensional space Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...

, a

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

five-dimensional space A five-dimensional (5D) space is a mathematical or physical concept referring to a space (mathematics), space that has five independent dimensions. In physics and geometry, such a space extends the familiar three spatial dimensions plus time ( ...

, 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." In a computer drawing application, the hidden line option may be referred to as a

wire frame Wireframe or wire-frame may refer to: * Wire-frame model, visual model of a three-dimensional object in computer graphics * Website wireframe, a basic visual guide used in web design See also

* Wire sculpture, used in plastic arts {{Disambig ...

with hidden edges. The wireframe hidden edges would be dashed lines. Obj lineremoval

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." Perspective Drawing of Surfaces with Hidden Line Removal(PDF English translation)
/ref> Point of View Mapping of a 3D Surface to 2D

Point of View Mapping of a 3D Surface to 2D

References

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External links

What are the advantages and disadvantages of using dashed or dotted hidden lines?
Elementary geometry Multi-dimensional geometry Graphical projections Technical drawing