Math Problems Vault

Problem 915

(1)(2)

In each of the following cases, find the projective transformation carrying the points $A, B, C$ into the points $A^{\prime }, B^{\prime }, C^{\prime }$ respectively:

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(1)

$A = A^{\prime }(0), B = B^{\prime }(1), C(2), C^{\prime }(\infty )$ ;

(2)

$A = A^{\prime }(0), B = C^{\prime }(1), C = B^{\prime }(2)$ .

Problem 916

(1)(2)(3)

(Types of projective transformations of a line). Under what necessary and sufficient condition does the projective transformation of the projective line

\lambda x_{1}^{\prime } = c_{11} x_{1}+c_{12} x_{2}, \quad \lambda x_{2}^{\prime } = c_{21} x_{1}+c_{22} x_{2}

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(1)

have no fixed points (an elliptic transformation);

(2)

have exactly two fixed points (a hyperbolic transformation);

(3)

have only one fixed point (a parabolic transformation)?

Problem 917

Prove that the projective transformation

\lambda x_{1}^{\prime } = 3 x_{1}-5 x_{2}, \quad \lambda x_{2}^{\prime } = x_{1}+x_{2}

is elliptic.

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Problem 918

(1)(2)(3)

Can a projective transformation of a line be improper if it is:

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(1)

elliptic;

(2)

hyperbolic;

(3)

parabolic?

Problem 919

Find the general form of the transformation leaving fixed the fundamental points $E_{1}$ and $E_{2}$ .

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Problem 920

Find the fixed points of the projective transformation

\lambda x_{1}^{\prime } = x_{1}+2 x_{2}, \quad \lambda x_{2}^{\prime } = 4 x_{1}+3 x_{2}

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Problem 921

Find all projective transformations under which two distinct points $(a: b)$ and $(c: d)$ are fixed.

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Problem 922

(Reflection with respect to a pair of points). Find all projective transformations of a line whose square equals the identity transformation, and determine their type.

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Problem 923

Three distinct points $A, B, C$ are given on a projective line. Prove that the projective transformation carrying these points respectively into the points $B, C, A$ is elliptic.

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Problem 924

A projective transformation of a line carries the points $A$ , $B, C$ into the points $A, C, B$ respectively. Using a straightedge, construct the second fixed point of this transformation.

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Problem 925

How many parabolic projective transformations of a projective line are there under which the given points $A$ and $B$ go to the given points $A^{\prime }$ and $B^{\prime }$ , provided that at least three of the four points $A, B, A^{\prime }, B^{\prime }$ are distinct?

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Problem 926

Let $f$ be an elliptic projective transformation of a line, and let $f^{k} = \underbrace{f \circ \ldots \circ f}_{k}$ be its $k$ -th power. Prove that if the transformation $f^{k}$ has a fixed point, then it is the identity.

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