Perspective collinear image of a circle
What is a perspective collinear image of a conic?
A conic is a set of ∞1 continuously connected points lying in a plane that each line intersects at two of its points. While, the property of continuously connected points is preserved under the perspective collineation, the image of a conic will again be some curve. Furthermore, each line in the plane of the curve is a perspective collinear image of a line in the conic plane which intersects the conic in two points (real and different, real and coinciding, conjugate imaginary). Since the collinearity is an invariance of the perspective collineation, these intersections will be preserved and we can conclude that the perspective collinear image of a conic is again a conic.
What is a perspective collinear image of a circle? It is a conic and its type will depend on its intersections with the line at infinity. Since the line at infinity is a perspective collinear image of the neutral line we can conclude the following:
If a circle does not intersect at real points the neutral line, then its perspective collinear image is an ellipse.
If a circle touches the neutral line, then its perspective collinear image is a parabola.
If a circle intersects the neutral line, then its perspective collinear image is a hyperbola.
Perspective collineation (S, o, C1, C2) and circle c1 with the center C1 are given on the figure 24. Perform the steps outlined in the sequence in the figure and study the corresponding comments.
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Figure 24
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1st step:
The perspective collinear image of the circle c1 is determined by constructing the images of the points lying on the circle. The construction of the point T2, the image of the point T1, is given on the figure.
Move the point T1 counterclockwise. Hence, the given circle is mapped to an ellipse under the given perspective collineation.
2nd step:
The circle tangent c1 at the point T1 is mapped onto a tangent of the obtained ellipse c2 with the tangent point at T2. Why?
Move the point T1 and notice that the statement is true for every pair of corresponding points of the circle and ellipse.
3rd step:
By moving the point A1 you can change the radius of the circle. Notice that two intersections points of the circle and its image are incident with the collineation axis, even if you change the radius of the circle.
Is the conic c2 always an ellipse?
4th step:
The line i1 is the neutral line of the given perspective collineation. Move the point A1 and check the statements above.
Place the circle c1 in such a position that its perspective collinear image c2 is a parabola.
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Since the perspective collineation does not preserve parallelism and midpoints, the diameter and center of a circle will NOT be mapped into diameter and center of its perspective collinear image.
Created by Sonja Gorjanc and Ema Jurkin, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb, made with GeoGebra
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