Introduction to Projective Geometry Solutions

An index to the solutions

A 3 minute read, posted on 27 Jul 2019
Last modified on 13 Aug 2019

Tags computer vision, projective geometry, problem solution

Although the author C. R. Wylie Jr. provides the solutions to most odd-numbered exercises, they are succinct to the point of being terse, and despite being a well-written enjoyable book1, it does not seem very popular with the computer vision or the mathematical community at the moment, resulting in a arduous path for those who want to study the subject on their own using this wonderful book as their guide. For these reasons, I’m documenting my answers to the exercises in this book.

I strongly recommend that you try to attempt the exercises yourself before looking at the solutions. Even if you initially feel that you’re making no progress, the mind is a strange and wonderful instrument and you might just get the answer after a restful nap.

My answers are organized into sections just like in the book. If you’re satisfied with your attempt and feel that you can’t proceed without help or you just want to confirm your answer then take a look!

Chapter 1: The Elements of Perspective

As the focus of this chapter is the synthetic approach to learning projective geometry, most problems are solved through construction. I started off doing the constructions by hand but thankfully discovered GeoGebra soon after and have been using it since. As the applet itself takes a while to load, I’ve uploaded the static image for each solution and linked to the applet in case you want to experiment with the solution. Click on the image to view it in full size. The $x$, $y$ and $z$ axes are colored red, green and blue respectively. Another really helpful tool is Maxima. Thankfully I don’t have to solve huge equations by hand anymore!

1.2 The Elements of Perspective

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1.3 Plane Perspective

I love coming back to these questions after working through the later chapters. This way I can discover the relationships the author was trying to highlight through the exercises. In this section we discuss plane perspective transformations which are also called perspectivities (Section 4.4). By considering images of images we also come across the concept of projectivities (Section 4.5) without calling it so. To be more specific, we are discuss collineations of type III (Section 5.6) in this chapter.

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Chapter 5: Linear Transformations in $\Pi_2$

5.2 Coordinate Transformations

This is mostly similar to the concept of “Change of basis”2 in Linear Algebra along with some particulars about the relationship between simultaneous point and line coordinate transformations.

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References


  1. C.R. Wylie, Jr. (1970), Introduction to Projective Geometry, Dover. [return]
  2. Wikipedia. Change of basis. https://en.wikipedia.org/wiki/Change_of_basis [return]
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