BAA013 Constructive Geometry

Credit conditions

To obtain credit:

• Unexcused absenses are not allowed.
• You need to score at least 40% in the sum of points for both tests.
• You need to submit 10 correctly drawn exercises.

Materials for lessons

• Materials for lesson 1:
• Ellipse
• Materials for lesson 2:
• Collineation and Affinity
• Materials for lesson 3:
• Monge projection
• Exercises
• Materials for lesson 6:
• Axonometry-theory
• Axonometry-exercises
• Materials for lessons 7-8:
• Plane sections of solids - theory
• Plane sections of solids - exercises
• The intersection of a line with a solid.
• Materials for lesson 9:
• Perspective projection

Exercises to submit

General information:

• draw exercises with a pencil on office paper, each exercise on a separate sheet
• mark them with your name, surname and date of issue
• submit exercises continuously (the recommended submission date is part of the assignment), you will avoid complications at the end of the semester

The assignment of exercises to submit:

• Homework 1 (submission deadline 26.2.2025): collinear image of a hexagon
• Homework 2 (submission deadline 5.3.2025): affine image of a pentagon
• Construction of a regular pentagon.
• Homework 3 (submission deadline 5.3.2025): affine image of a circle
• Homework 4 (submission deadline 12.3.2025): In MP: construct related views of the square ABCD lying in the plane alpha(60,60,45), A[-25,?,45], C[-15,65,?].
• Homework 5, 6 (submission deadline 2.4.2025): Construction of solids in MP
• Homework 7 (submission deadline 9.4.2025): Construction of a solid in OA
• Homework 8 (submission deadline 16.4.2025): Plane sections of solids in OA
• Homework 9 (submission deadline 23.4.2025): The intersection of a solid and a line in OA
• Homework 10 (submission deadline 30.4.2025): Perspective projection

Exercises to practice

• These exercises do not need to be submitted, they serve as preparation fot the tests. They are intended for practicing.

• Exercise from lesson 1: the ellipse is given by major vertices and by its point M, construct tangents to this ellipse parallel to the given direction s and construct tangents to the ellipse passing through the point R (is not at the ellipse)
• Exercises from lesson 3 Affinity
• Monge projection Exercises from lesson 3 and Exercises from lesson 4
• Exercises from lesson 6
• Exercises from lesson 8
• The intersection of a line with a cylinder.
• The intersection of a line with a cone.
• Exercises from lesson 9 and 10: Perspective projection

Exercises discussed in lessons

Lesson 1 Ellipse

• Introduction to the conic sections

• Point construction, a tangent, osculating circles here.
• Vertex circle and Directrix circle here.
• Tangent to the ellipse parallel to given direction here.
• More exercises to practise here.

• Ellipse - Tangent to the ellipse passing through the outer point here.
• Ellipse - Trammel construction here.
• Collineation – exercise 1, exercise 2, exercise 3, exercise 4.

• Affinity – exercise 1, exercise 2.
• Affine image of a circle:
  – in the orthogonal affinity here.
  – in an oblique affinity – using general conjugate diameters here.
  – Rytz construction here.
• More exercises to practise collineation, affinity.

• Trace points of a line, the true length of the line segment exercise 1
• Traces of the plane exercise 2
• Basic problem IIb: a point of intersection of a straight line with a plane here
• Basic problem IIIa: a line perpendicular to a plane here
• Horizontal and vertical (frontal) lines, lines of steepest slope here
• More exercises to practise here.

• Distance of a point from a plane exercise 3.
• Basic problem IIIb: a plane perpendicular to a line here
• Distance of a point from a line exercise 4.
• Rotation of the point lying in the plane here
• Equilateral triangle here.

• Exercises - construction of solids in MP.
• Projection of pyramid.
• Related views of a circle given by its center and its one point here
• Projection of the cylinder in MP.

• Test 1 (50 min)

• Axonometry
• Trace points of a straight line.
• Traces of a plane given by parallel lines.
• The intersection of planes.
• The intersection of a straight line with a plane.

• The axonometric projections of a point and a plane given by coordinates.
• Regular pyramid with the base in the horizontal projection plane.
• The circle lying in the horizontal projection plane given by the center and one point.

• The circle lying in the horizontal projection plane given by the center and the radius.


• Plane sections of solids - theory
• Plane sections of solids - exercises
• The plane section of a prism.
• The plane section of a pyramid.
• The plane section of a right circular cylinder.


• The intersection of a line with a solid.
• The intersection of a line with a prism.
• The intersection of a line with a pyramid.


• More exercise to practise in the collection.

• Perspective projection
• Basic facts.
• Perspective projections of straight lines.
• The angle of intersecting lines.
• Reminder of basic constructions known from the lecture:
  • The length of the segment on line in the ground plane in general position - by using measuring points.
  • The length of the segment on depth lines.
  • The length of the segment on lines paralel to the picture plane.
  • The height - on vertical lines.
  • Connect point B with an inaccessible vanishing point.
• Perspective projection of a rectangle in the ground plane.
• Construction - inaccessible vanishing point.
• Regular quadrilateral pyramid with the base in the ground plane.
• Regular quadrilateral pyramid with the base in the ground plane.

• Perspective projection of crossroad
• Similar example.
• Regular quadrilateral prism - the tipped ground plan method.
• Equilateral triangle in the ground plane– the tipped ground plan method - reduced distance.

• Test 2 (50 min)


• Perspective projection
• Perspective projection of stairs
• Similar example.
• The exercise recommended for practicing the tipped ground plan method: the object given by its tipped ground plan.

• Recommended exercise:
• Right circular cylinder.
• Perspective projection of circle in the ground plane.
• Perspective projection of canopy
• Perspective projection of circle in the plane that is perpendicular to the ground plane.
• Similar example.

• More exercise to practise in the collection.
• Methods for the construction of a ground plan.

Tests

The maximum of the first test is 15 points. The maximum of the second test is 20 points. To obtain credit you need to score at least 40% (14 points of 35 points) in the sum of points for both tests. There will be an opportunity to do two make-up tests - in the examination period.

• Test 1 (50 min) - 19.3.2025
• Tasks that can appear on the first test:
  • Tangents to the ellipse parallel to the given direction, tangents to the ellipse passing through the outer point, construction of the ellipse using focal properties
  • The affine image of a square or a triangle, the affine image of a circle, the image of a square in the collineation
  • Monge projection:
    • Basic tasks concerning a plane (a point and a straight line in a plane, traces)
    • Basic problems
    • Distance of a point from a plane
    • Distance of a point from a line
    • Related views of an object in a plane (a square, a triangle, a circle)
• Sample first test

• Test 2 (50 min) - 23.4.2025
• Tasks that can appear on the second test:
  • Orthogonal axonometry:
    • Axonometric view of a point.
    • Axonometric view of an object in a first projection plane (a square, a triangle, a circle).
    • Plane section of prism or pyramid with the base in a first projection plane.
    • The intersection of a line with a solid.
  • Perspective projection
    • Construction in the ground plane (lengths, parallels, the angle between intersecting lines...)
    • Perspective projection of a triangle or a square in the ground plane.
    • The tipped ground plan method.
    • Construction - an inaccessible vanishing point, reduced distance.
    • Perspective projection of a solid with the base in the ground plane.
• Sample second test