| |
| |
Caption: Mathematical Models by Arnold Emch - Series 4 (1928)
This is a reduced-resolution page image for fast online browsing.
EXTRACTED TEXT FROM PAGE:
IV Series 33 This constructive treatment of perspective is especially effective for the projection of infinite singularities. For example it can be made graphically plausible that the infinite point of the cubical parabola y = x3 is a cusp, or that the infinite point of the semi-cubical parabola y2 = x3 is a flex. The construction becomes particularly simple when a = b. The perspective is now involutorial and the characteristic cross-ratio is — 1 . The model has an aluminum base for 2 and a glass-plate for 2' before i is made to coincide with 2. t 48. A fine Transformation Between two Planes. The general affine relation between two planes 2' (V, y') and 2 (x, y) is given by xf = ax + by + c y' = dx + ey + f with the matrix a b c of rank 2. def In case of an affine homology, or perspective, there exists a pointwise invariant line. If we choose this as the x-axis, then for y = 0, there must be x' = x, y' — 0. This is only possible when x' = x + by, yr = ey. The joins of corresponding points are parallel and have the slope (y' — y)/(xf — x) = (e — l)/&. If we denote two corresponding points by P and P' and the intersection of their join with the x-axis by S, then, as in perspective, the cross-ratio (P'PS oo) = P'S/PS = y'/y = e is constant. In the model the two planes 2 and 2r are hinged along the xaxis and m a y be turned about it. The points of a circle in 2 and its affinefigure,an ellipse, in 2r are joined by threads which slip through the corresponding points, the threads, are the generatrices of a variable elliptical cylinder. 2 and 2r are represented by aluminiumplates, Fig. 21. 49. The Space Sextic of Genus 4. Models for sextics of this type are described in Series II, pp. 8-12. The object of these models is to show the six double-points of the projection of the sextic from a generic point in space upon a generic plane.
| |
