20 Projective Geometry Quiz Questions and Answers

Question 1:
In projective geometry, what is the fundamental theorem that states every collineation is a projective transformation?
A. Desargues’ theorem
B. Pascal’s theorem
C. Pappus’s theorem
D. The fundamental theorem of projective geometry
Answer: D
Explanation: The fundamental theorem of projective geometry states that every collineation between two projective spaces is a projective transformation, meaning it preserves the cross-ratio and incidence relations.

Question 2:
What is the cross-ratio of four collinear points in projective geometry?
A. A ratio that remains invariant under projection
B. The distance between the first and last point
C. The sum of the distances between pairs of points
D. A measure of angle between lines
Answer: A
Explanation: The cross-ratio is a fundamental invariant in projective geometry, defined for four collinear points, and it remains unchanged under projective transformations.

Question 3:
In projective geometry, how many points at infinity are there on a line in the real projective plane?
A. One
B. Two
C. Infinite
D. None
Answer: A
Explanation: In the real projective plane, every line intersects the line at infinity at exactly one point, representing the direction of the line.

Question 4:
What is a conic section in projective geometry?
A. A curve defined by a second-degree equation
B. A straight line
C. A circle only
D. A set of parallel lines
Answer: A
Explanation: A conic section is the locus of points satisfying a second-degree equation in homogeneous coordinates, which can represent ellipses, parabolas, hyperbolas, or degenerate cases.

Question 5:
Which theorem states that two triangles are perspective from a point if and only if they are perspective from a line?
A. Desargues’ theorem
B. Ceva’s theorem
C. Brianchon’s theorem
D. Cayley’s theorem
Answer: A
Explanation: Desargues’ theorem establishes a duality between points and lines, stating the condition for two triangles to be perspective.

Question 6:
In projective geometry, what is the dual of a point?
A. A line
B. A plane
C. Another point
D. A conic
Answer: A
Explanation: Duality in projective geometry swaps points and lines, so the dual of a point is a line, preserving incidence relations.

Question 7:
What is the projective plane defined as?
A. A plane plus a line at infinity
B. A set of points and lines with incidence relations
C. Only the Euclidean plane
D. A sphere
Answer: B
Explanation: The projective plane is an abstract space where points and lines satisfy certain axioms, including the addition of points at infinity.

Question 8:
How does a projective transformation affect parallel lines?
A. They intersect at a point at infinity
B. They remain parallel
C. They become perpendicular
D. They disappear
Answer: A
Explanation: Projective transformations map parallel lines to lines that intersect at a point at infinity in the projective plane.

Question 9:
What is the order of a projective transformation in the projective line?
A. It can be represented by a 2×2 matrix up to scalar multiple
B. It is always linear
C. It involves only translations
D. It is defined only for three dimensions
Answer: A
Explanation: Projective transformations on the projective line are induced by invertible linear transformations in homogeneous coordinates, represented by 2×2 matrices modulo scalars.

Question 10:
In projective geometry, what is a pencil of lines?
A. A set of lines passing through a common point
B. A set of parallel lines
C. A conic section
D. A pair of intersecting planes
Answer: A
Explanation: A pencil of lines is the collection of all lines that pass through a fixed point, forming a one-parameter family.

Question 11:
Which of the following is invariant under projective transformations?
A. Cross-ratio
B. Euclidean distance
C. Angle measure
D. Area
Answer: A
Explanation: The cross-ratio is the only invariant preserved under projective transformations, while distances, angles, and areas are not.

Question 12:
What is the dimension of the projective space RP^n?
A. n
B. 2n
C. n+1
D. Infinite
Answer: A
Explanation: The real projective space RP^n is an n-dimensional manifold, obtained by quotienting R^{n+1} minus the origin by scalar multiplication.

Question 13:
In projective geometry, how are two conics related if they are tangent?
A. They intersect at exactly one point with the same tangent
B. They do not intersect
C. They are identical
D. They intersect at two points
Answer: A
Explanation: Two conics are tangent if they intersect at a point and share the same tangent line at that point.

Question 14:
What is Brianchon’s theorem?
A. For a hexagon circumscribed about a conic, the main diagonals are concurrent
B. For a triangle inscribed in a conic, the sides are concurrent
C. It relates to parallel lines
D. It defines the cross-ratio
Answer: A
Explanation: Brianchon’s theorem is the dual of Pascal’s theorem, stating that in a hexagon circumscribed around a conic, the lines joining opposite vertices are concurrent.

Question 15:
In homogeneous coordinates, how is a point represented?
A. As a tuple (x, y, z) not all zero, up to scalar multiple
B. As Cartesian coordinates only
C. As a vector in Euclidean space
D. As a single number
Answer: A
Explanation: Homogeneous coordinates represent points in projective space as equivalence classes of vectors, where scalar multiples are identified.

Question 16:
What is the significance of the line at infinity in projective geometry?
A. It compactifies the plane and handles parallel lines
B. It separates points
C. It is not part of the plane
D. It defines distances
Answer: A
Explanation: The line at infinity adds points to the Euclidean plane, making it compact and ensuring that parallel lines intersect there.

Question 17:
Which property distinguishes projective geometry from Euclidean geometry?
A. Preservation of incidence rather than distance
B. Measurement of angles
C. Use of metrics
D. Straight lines only
Answer: A
Explanation: Projective geometry focuses on incidence relations (points on lines) rather than metrics like distance or angles, which are central in Euclidean geometry.

Question 18:
What is a projective collineation?
A. A bijection that preserves collinearity
B. A transformation that preserves distances
C. A rotation
D. A translation
Answer: A
Explanation: A projective collineation is a transformation that maps collinear points to collinear points, maintaining the structure of lines.

Question 19:
In projective geometry, how many lines pass through two distinct points?
A. Exactly one
B. Two
C. None
D. Infinite
Answer: A
Explanation: By the axioms of projective geometry, any two distinct points determine a unique line.

Question 20:
What is Pascal’s theorem?
A. For a hexagon inscribed in a conic, the intersection points of opposite sides are collinear
B. For two lines, they are parallel
C. It relates to triangles only
D. It defines points at infinity
Answer: A
Explanation: Pascal’s theorem states that if a hexagon is inscribed in a conic section, then the three points of intersection of its opposite sides are collinear.