Projective Geometry
Geometry with points at infinity; perspective and duality.
Key idea
Geometry with points at infinity; perspective and duality.
In depth
Projective geometry adds points at infinity so that any two lines meet. Properties preserved by projection, like collinearity and cross-ratio, become the focus. It underpins perspective drawing and computer vision.
Worked example
Parallel railway lines appear to meet at a vanishing point.
Real-world uses
- Perspective drawing in art and the maths behind computer vision and camera projection
Common mistake
Expecting distances and parallelism to be preserved by projection.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Projective Geometry?
Geometry with points at infinity; perspective and duality. It is a topic at the expert level in Geometry.
Where is Projective Geometry used in real life?
Real-world examples: Perspective drawing in art and the maths behind computer vision and camera projection.
What should I learn before Projective Geometry?
Make sure you are comfortable with Spherical Geometry, the previous topic in Geometry.
What is a common mistake with Projective Geometry?
Expecting distances and parallelism to be preserved by projection.
How can I practise Projective Geometry?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Projective Geometry?
Move on to Topology of Surfaces (Intro) next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.