Perspective collinear image of a polygon
Due to the properties of the perspective collinear image of a line segment, we can conclude that the perspective collinear image of a polygon will be a polygon only in the case when the given polygon is not intersected by the neutral line of the given perspective collineation. Only then, the given polygon will be mapped onto a polygon with the same number of sides and vertices. But even in these cases a regular polygon will not be mapped onto a regular polygon because the perspective collineation does not preserve the distance between points.
If there is a point of the polygon lying on the neutral line, then its perspective collinear image will not be a polygon.
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Slika 22
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Perspective collineation (S, o, A1, A2) and equilateral triangle Δ A1B1C1 are given on the figure 22.
Construct its perspective collinear image and then explore what forms its image takes by moving the position of the triangle Δ A1B1C1 is moved.
For the given position of the triangle Δ A1B1C1 the perspective collinear image is the scalene triangle Δ A2B2C2. This happens because the neutral line does not intersect the triangle Δ A1B1C1. Check this statement.
By moving points H and A1 the triangle Δ A1B1C1 can be placed into different positions with respect to the neutral line i1 so that its perspective collinear image will not be a triangle. There are 4 such positions:
- i1 is incident with only one triangle vertex,
- i1 is incident with one triangle vertex and it intersects one triangle side,
- i1 intersects two triangle sides,
- i1 lies on one triangle side.
Place the triangle Δ A1B1C1 in each of these position and note what is its perspective collinear image.
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Figure 23
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Perspective collineation (S, o, A1, A2) and square A1B1C1D1 are given on the figure 23.
Construct its perspective collinear image, explore its properties related to the parallelism of the opposite sides of the square and investigate which forms its image takes by moving the position of the square.
For a given position of a square A1B1C1D1, generally, the perspective collinear image is some quadrilateral A2B2C2D2 that has no parallel sides. The lines, in which the images of the parallel sides lie, intersect at a point on the neutral line.
By moving the points O and A1 you can place the given square into 4 different positions with respect to the neutral line i1 such that its image will not be a quadrilateral.
Place the square A1B1C1D1 into each of these positions and notice what is its perspective collinear image.
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Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb, made with GeoGebra
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