Math 7: Special Projectivities in Geometry
40 Questions
2 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

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.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

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

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser