Descriptive geometry

Builds on Euclidean space Plane Cartesian coordinate system Perspective projection

Definition

Let $E$ be Euclidean 3-space with Cartesian coordinates $(x, y, z)$ . Let $\pi_1$ be the horizontal plane $z = 0$ and $\pi_2$ the vertical plane $y = 0$ ; they meet in the ground line $x_{12}$ , the $x$ -axis. The Monge projection of a point $P = (x, y, z)$ is the pair of its orthogonal projections, the top view $P' = (x, y, 0) \in \pi_1$ and the front view $P'' = (x, 0, z) \in \pi_2$ . Rotate $\pi_2$ about $x_{12}$ onto $\pi_1$ , so that both views lie in one drawing plane. Then $P'$ and $P''$ lie on a common line perpendicular to $x_{12}$ . Descriptive geometry is the study of figures of $E$ through such pairs of views, or through more views on further image planes. Its aim is to solve spatial problems by plane constructions on the views.

Remarks

The map $P \mapsto (P', P'')$ is injective, since the two views give $x$ , $y$ and $z$ . So a pair of views on a common perpendicular to $x_{12}$ determines a unique point. Example: for a segment $PQ$ the true length follows from the views, since $\abs{PQ}^2 = \abs{P'Q'}^2 + (z_P - z_Q)^2$ , where $\abs{P'Q'}$ is measured in the top view and $z_P - z_Q$ in the front view. For $P = (0, 0, 0)$ and $Q = (3, 4, 12)$ , one gets $\abs{P'Q'} = 5$ and $\abs{PQ} = \sqrt{25 + 144} = 13$ . Non-example: a single view does not determine the point: the points $(1, 2, 0)$ and $(1, 2, 5)$ have the same top view. The Euclidean space can also be shown by central projections, as in a perspective projection.

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Tag	`8WETP2V4MB`
URI	`m2://8WETP2V4MB`
Kind	definition
Writing	W0 stub
MSC	51N05
Defines	Descriptive geometry
Uses	Euclidean space, Plane, Cartesian coordinate system, Perspective projection

Descriptive geometry

Definition

Remarks

Reading path

Further reading

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