Math 7: Special Projectivities in Geometry

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Questions and Answers

What is the number of invariant points for an elliptic projectivity?

  • Three
  • Zero (correct)
  • Two
  • One

What is the condition for a projectivity to be the identity transformation?

  • It has three invariant points (correct)
  • It has one invariant point
  • It has two invariant points
  • It has four invariant points

What is uniquely determined when both invariant points and one other set of corresponding points are given?

  • A parabolic projectivity
  • Any projectivity
  • An elliptic projectivity
  • A hyperbolic projectivity (correct)

What is the condition for a parabolic projectivity to be uniquely determined?

<p>When its invariant point and one other set of corresponding points are given (A)</p> Signup and view all the answers

What is the geometric object that intersects with the given line at point D?

<p>Line SS' (A)</p> Signup and view all the answers

What is the condition for a projectivity to be parabolic?

<p>If D is equal to A (A)</p> Signup and view all the answers

What is the harmonic conjugate of B with respect to A and B'?

<p>B'' (A)</p> Signup and view all the answers

What is the center of the two perspectivities?

<p>Point S (A)</p> Signup and view all the answers

What is the result of the matrix transformation that causes lines parallel to the z-axis to meet at the vanishing point (0, 0, 1/3 , 1)?

<p>Lines parallel to the z-axis meet at the vanishing point (0, 0, 1/3 , 1) (B)</p> Signup and view all the answers

What type of transformation is represented by the matrix that results in a projection onto the plane z = 2?

<p>Projection transformation (C)</p> Signup and view all the answers

What is the determinant of the matrix for an axonometric projection?

<p>Zero (C)</p> Signup and view all the answers

What is a projectivity of period n?

<p>A projectivity that must be repeated n times before it first results in the identity transformation. (D)</p> Signup and view all the answers

What is the purpose of axonometric projections in engineering?

<p>To produce various two-dimensional views of a three-dimensional object (B)</p> Signup and view all the answers

What is an involution?

<p>A projectivity of period 2. (B)</p> Signup and view all the answers

What is the result of the matrix transformation that results in a rotation of 90° about the x-axis, then a projection onto the plane z = 0?

<p>Rotation of 90° about the x-axis, then a projection onto the plane z = 0 (B)</p> Signup and view all the answers

What type of perspectives are examples of matrices for shown in Figure 4.15b-d?

<p>All of the above (D)</p> Signup and view all the answers

According to Theorem 4.5.2, what is the minimum condition for a projectivity to be an involution?

<p>The projectivity must interchange one pair of distinct points. (A)</p> Signup and view all the answers

What is the effect of a one-dimensional involution on a pair of points?

<p>It interchanges the pair of points. (A)</p> Signup and view all the answers

What is the characteristic of the transformation matrices to transform a box into a figure with pairs of edges meeting at one or more vanishing points?

<p>One or more nonzero elements in submatrix C (D)</p> Signup and view all the answers

What is the result of an axonometric projection onto the plane z = 2?

<p>All image points are on the plane z = 2 (D)</p> Signup and view all the answers

What is the effect of a two-dimensional projectivity on a line in the plane?

<p>It transforms the line into another line. (C)</p> Signup and view all the answers

What is the condition for a two-dimensional projective transformation to be the identity transformation?

<p>It leaves four lines invariant. (B)</p> Signup and view all the answers

What is the geometric object that is transformed by a two-dimensional projectivity?

<p>A one-dimensional set. (B)</p> Signup and view all the answers

What is the effect of a two-dimensional projectivity on the points of a line?

<p>It transforms the points to new points. (B)</p> Signup and view all the answers

What is the condition for a projectivity to be equivalent to a perspectivity?

<p>The projectivity has a self-corresponding line (A)</p> Signup and view all the answers

What is the role of the axis of projectivity in a projectivity?

<p>It is used to construct additional pairs of corresponding points (C)</p> Signup and view all the answers

What is the significance of the point of intersection of 𝐴𝐵′ and 𝐴′𝐶 in the given projectivity?

<p>It is the axis of projectivity (A)</p> Signup and view all the answers

What is the condition for two pairs of points to have the same axis of projectivity?

<p>The points are images of the common point of the two lines (C)</p> Signup and view all the answers

What is the role of the cross joins in the given projectivity?

<p>They meet at the image of the common point of the two lines (D)</p> Signup and view all the answers

What is the use of the given pairs of corresponding points in a projectivity?

<p>To construct additional pairs of corresponding points (C)</p> Signup and view all the answers

What is the significance of the point A'A in the given projectivity?

<p>It is a self-corresponding line (A)</p> Signup and view all the answers

What is the relationship between the projectivity and the perspectivity?

<p>The projectivity is equivalent to the perspectivity (A)</p> Signup and view all the answers

What is the consequence of Theorems 4.4.2 through 4.4.4?

<p>They provide an alternative definition of a projective transformation of a plane onto itself. (B)</p> Signup and view all the answers

What is the property preserved by a projective transformation according to Theorem 4.4.3?

<p>Cross ratio of four points (D)</p> Signup and view all the answers

What is the condition required for the equations of a projective transformation in the projective plane?

<p>The determinant of the coefficients is not zero (D)</p> Signup and view all the answers

What is the transformation that preserves the cross ratio of every four collinear points?

<p>A projective transformation (D)</p> Signup and view all the answers

What is the type of transformation that preserves collinearity?

<p>A projective transformation (A)</p> Signup and view all the answers

What is the form of the equations of a projective transformation in the projective plane?

<p>x' = ax + by + cz, y' = dx + ey + fz, z' = gx + hy + kz (A)</p> Signup and view all the answers

What is the cross ratio of the four points (0, 0, 1), (0, 1, 1), (0, 1, 0), and (0, x2, x3)?

<p>x2/x3 (A)</p> Signup and view all the answers

What is the significance of the four points (0, 0, 1), (0, 1, 1), (0, 1, 0), and (0, x2, x3)?

<p>They represent any four points on the line x1 = 0 (C)</p> Signup and view all the answers

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Study Notes

One-Dimensional Projectivities

  • A one-dimensional projectivity is a transformation that renames points on the same line.
  • A one-dimensional projectivity can be classified into three types: elliptic, parabolic, or hyperbolic, depending on the number of invariant points (zero, one, or two, respectively).
  • If there are three invariant points, the projectivity is the identity transformation.

Hyperbolic Projectivity

  • A hyperbolic projectivity is uniquely determined when both invariant points and one other set of corresponding points are given.

Parabolic Projectivity

  • A parabolic projectivity is uniquely determined when its invariant point and one other set of corresponding points are given.
  • A parabolic projectivity can be expressed as a product of two perspectivities with a center and a line chosen arbitrarily.

Two-Dimensional Projectivities

  • A two-dimensional projectivity transforms every one-dimensional set projectively.
  • A two-dimensional projective transformation may involve every point of the plane, but each line in the plane is transformed into another line, establishing a projectivity by the points on the two lines.
  • A two-dimensional projective transformation that leaves the four lines of a complete quadrilateral invariant is the identity transformation.

Projective Transformations

  • A projective transformation is a one-to-one mapping of a plane onto itself that preserves collinearity and the cross ratio of points.
  • A projective transformation can be represented by a matrix equation, with the stipulation that the determinant of the coefficients is not zero.

Periodic Projectivities

  • A projectivity of period n is one that must be repeated n times before it first results in the identity transformation.
  • An involution is a projectivity of period 2, which interchanges pairs of points.
  • A one-dimensional projectivity that exchanges one pair of distinct points is an involution.

Axonometric Projections

  • Axonometric projections are used to produce various two-dimensional views of a three-dimensional object.
  • Axonometric projections technically are mappings rather than transformations, and the determinant of the matrix is zero, so there is no inverse.
  • The matrix for an axonometric projection can be considered the product of two other matrices: one for rotation or translation, and another for projection onto a particular plane.

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