What is Technical Drawing, Descriptive Geometry, Projective Geometry, Linear Algebra

Xah Lee, 2006-11, 2008-01

[This essay discusses the relations between Technical Drawing, Descriptive Geometry, Projective Geometry, Linear Algebra, and CAD.]

Technical Drawing and Descriptive Geometry

In the early 1990s, when i was in my early twenties, i was fascinated by Technical drawing↗. Technical drawing is those you see in wire frame↗ cars, architecture blue-prints, mechanical devices designs, etc. This skill fascinated me. In college, you can major in technical drawing and become a draftsman.

above: A technical drawing. (done with modern CAD software) (Source↗ 2008-03)

Technical drawing is a skill, a vocation, but today it is much replaced by operating Computer-aided design↗ software, e.g. AutoCAD↗ and far numerous other 3D-modeling software today. (See 3D modelers↗)

The theories behind technical drawing, is called Descriptive Geometry↗.

Let's say, given a cube. And you are told how it is positioned on the table, and where you are viewing it from with a camera. Could you, sketch a drawing, so that, when placed on the photo edge-to-edge, the cube's corners match exactly?

Descriptive geometry, is the engineering discipline that helps you determine the answer. (so that you can draw photo-realistic pictures.)

However, note that “descriptive geometry” is not a branch or methodology of geometry, despite its name ending in “geometry”. It is more of a engineering discipline. Because, it is mostly concerned about drawing realistic pictures based on the principles of basic Euclidean geometry↗. It is not concerned about geometric theorems.

Projective Geometry

Now, the gist of technical drawing is just linear projection. That is, project points in 3D space into a 2D paper, in such a way it is realistic. This is basically how our eye works. Light from 3D space come into our pupil and is projected back onto our 2D retina.

The theoretical aspects in projection, is known as projective geometry↗. It is concerned about, for example, what properties are invariant under projection. Projective geometry, is indeed, a major branch of math. However, due to the nature of projective geometry, it has no practical connection to technical drawing.

above: Desargues's Two Triangle Theorem.

For example, one of the most important theorem in projective geometry is called Desargues's Two Triangle Theorem. It basically states how any 2 triangles's sides and the corners, when connected in a certain way, always meet in a single point or on a single line. For more about the subject, see: Introduction to Real Projective Plane.

Linear Algebra

Another subject intimately related to descriptive geometry and projective geometry is linear algebra.

Before the computer days, when a technical drawer wants to draw something, he uses priciples in Descriptive Geometry, with tools like T-square↗ , compass, and drafting table↗. In modern days of computing, Descriptive Geometry is falling out, because we can now compute the exact coordinates of points in the scene and their projection on the plane, directly, using computer and linear algebra, and thus bypassing those indirect mathematical rules and principles of how to conduct drawing faithfully.

This direct computation of coordinates, is part of linear algebra. In particular, matrix multiplication. So, in CAD software, it is the matrix multiplication that is at play, with results that is more direct and mathematically perfectly accurate than the nature of descriptive geometry. Note that, this is only possible with massive computing power.

Summary

In summary:

• The nature of Technical drawing is drawings made by linear projection.
• Descriptive Geometry is a collection of guides and principles that help people on how to accurately do technical drawing. This is before the time of modern computers with CAD software.
• Projective Geometry is a branch of mathematics that studies some properties about projection. The motivation of projection geometry began when graphics artists in the 1600s discovered the technique of perspectivity that lets them create drawings that have a realistic, 3D sense (as opposed to being flat). However, Projective Geometry quickly developed into a pure, abstract mathematics, having absolutely nothing to do with drawings or its technique.
• Linear Algebra is a branch of mathematics that studies some properties that originates from linear polynomials. In particular, one of its main application is to determine the exact coordinates from linear projection (matrix multiplication). This is how CAD software is able to compute the coordinates of 3D models for displaying on the 2D screen.

Related essays:

• What Is Perspective Drawing

Page created: 2008-01.
© 2006 by Xah Lee.