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Math 7: Special Projectivities in Geometry
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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</p> Signup and view all the answers

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

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

    What is the condition for a projectivity to be parabolic?

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

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

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

    What is the center of the two perspectivities?

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

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

    <p>Zero</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.</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</p> Signup and view all the answers

    What is an involution?

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

    What is the type of transformation that preserves collinearity?

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

    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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    Explore special cases of projectivities in one and two dimensions, including elliptic, parabolic, and hyperbolic one-dimensional projectivities.

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