Geometry in architecture and building by Hans Sterk

By Hans Sterk

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Second, corresponding to this solution we find the vector p = (−1, 3, 2) and q = (1, 7, −2). The distance between the lines is therefore √ (1 − −1)2 + (7 − 3)2 + (−2 − 2)2 = 36 = 6. 1 In designing shapes, you may want to move or rotate certain elements of your design, say a window or a wall. The question is how you do this mathematically. Such mathematical computations are at the basis of computer software implementations. In this section, we will briefly discuss a few aspects of translations.

The patterns have various centers of symmetry. A rotation over a specific angle transforms the pattern into itself. 19: Rotational symmetry. All black dots are centers of symmetry, at least if you imagine the pattern to extend indefinitely. On the right the effect of a rotation over 180◦ on one of the rectangles is illustrated. c) Mirror symmetry or reflectional symmetry. Patterns may be transformed into themselves by reflecting them in suitable lines. In the picture a few of such lines are indicated.

C) Use the previous results to compute the distance between the planes. d) Computing the distance between two parallel planes is related to computing the distance between a point and a plane. Explain this relation. 21 The relative position of a plane and a sphere In Chapter 1, Ex. 6, the distance between the origin and the plane with equation √ x + y + z = a was given as |a|/ 3. a) Compute this distance by first intersecting the line (x, y, z) = λ(1, 1, 1) with the plane x + y + z = a, and then computing the distance between the origin and the point of intersection.

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