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

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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.

Wright brothers patent plans 1908
Wright brothers's Patent drawing of their glider, 1908. A typical technical drawing by hand. img src

Technical drawing is a skill, a vocation, but today it is much replaced by operating Computer-aided design (CAD) software. (for example, AutoCAD, SketchUp, and many others. See: List of 3D Modeling Software.)

Descriptive Geometry = Principles of Technical Drawing

The techniques, methods, principles, of technical drawing, is called Descriptive Geometry.

Albrecht Durer Man Drawing a Lute
〈Man Drawing a Lute〉, by Albrecht Dürer (1471 〜 1528). This drawing shows a simple but impractical way to correctly do a perfect drawing. The object to draw is the lute. The camera/eye position is the nail on the wall. The paper (shown as a door) is the plane the projection is to be onto.

Suppose there is a ashtray on a table. Now, you are to draw this ashtray. Could you, sketch a drawing, so that, when your drawing is placed on top of a photograph of it, they match exactly?

Descriptive geometry, is the engineering discipline that helps you draw pictures correctly.

The name “descriptive geometry” might seems to mean it's some study of geometry, but actually it is not a study of math or geometry. A better name would be “principles of technical drawing”.

drafting table
A classic drafting table, probably dated around 1950s. img src
Drafter at work
Modern drafter at work. img src
Descriptive geometry sample page
A typical textbook page on descriptive geometry. img src
technical drawing
A technical drawing. Drawn by modern CAD software. http://en.wikipedia.org/wiki/Image:Ss_sketchup30osx.JPG

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.

desarguesTheorem
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.

Linear Algebra, is a branch of math that has application in geometry. In our context, it lets you compute the exact coordinates of a point in space on a given plane with a given camera position.

Before our digital computer era, when a technical draftman wants to draw something, he uses principles in Descriptive Geometry, with tools like T-square , compass, and drafting table. In modern days of computing, Descriptive Geometry is becoming little used, 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 do drawing correctly.

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 perfectly accurate than the nature of descriptive geometry. Note that, this is only possible with massive computing power.

Summary

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