Question

Let \(P_{1}, P_{2}, P_{3}\) (resp. \(\left.Q_{1}, Q_{2}, Q_{3}\right)\) be three points in \(\mathbb{P}^{2}\) not lying on a line. Show that there is a projective change of coordinates \(T: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2}\) such that \(T\left(P_{i}\right)=Q_{i}\) \(i=1,2,3\). Extend this to \(n+1\) points in \(\mathbb{P}^{n}\), not lying on a hyperplane.

Short answer

In order to find a projective transformation that maps a given set of three points in the projective space $\mathbb{P}^2$ to another set of three points, follow these steps: 1. Understand the projective transformation (homography) can be represented as a non-singular $3\times3$ matrix that maps lines to lines in $\mathbb{P}^2$. 2. Assign each point from the first set to the corresponding point in the second set and set up a system of equations relating their homogeneous coordinates. 3. Determine the transformation matrix $M$ by solving the system of equations. 4. Extend the results to $\mathbb{P}^n$ by working with an $(n+1) \times (n+1)$ matrix and following similar steps as before. Remember, this transformation is only possible if the points do not lie on a line or hyperplane. If the points lie on a line or hyperplane, the transformation matrix will be singular, and no valid projective transformation will exist.

Step-by-step solution

Understand projective transformations

Projective transformations, or homographies, are transformations in the projective space that map lines to lines. In this particular problem, we are working in the projective space \(\mathbb{P}^2\). A projective transformation in \(\mathbb{P}^2\) can be represented by a non-singular \(3\times3\) matrix. The transformation can be written as \(T(p) = Mp\), where \(p\) is the vector of homogeneous coordinates of the point, and \(M\) is the matrix. In other words, for a given point \([x : y : z] \in \mathbb{P}^2\), the projective transformation is defined by \([x' : y' : z'] = [ax + by + cz : dx + ey + fz : gx + hy + iz]\), where \(a, b, c, d, e, f, g, h, i\) are constants.

Set up a system of equations

To find the projective transformation \(T\) such that \(T(P_i) = Q_i\), we can set up a system of equations relating their homogeneous coordinates. Let \(P_i = [x_i : y_i : z_i]\) and \(Q_i = [x_i' : y_i' : z_i']\). Then we have the following equations for \(i=1,2,3\): $$ \begin{cases} a x_i + b y_i + c z_i = x_i' \\ d x_i + e y_i + f z_i = y_i' \\ g x_i + h y_i + i z_i = z_i' \end{cases} $$ Keep in mind that since we are working in projective space, we are dealing with multiples of the coordinates.

Determine the transformation matrix

To determine the transformation matrix \(M\), we can solve the system of equations from step 2. There are various ways to solve the system of equations, such as substitution or elimination methods. The solution will give us the matrix \(M\), and consequently, the projective transformation \(T\). However, it's important to note that this transformation is only possible if the points do not lie on a line. If the points \(P_i\) or \(Q_i\) lie on a line, the transformation matrix M will be singular, and no valid projective transformation will exist.

Extend the result to \(\mathbb{P}^n\)

Now we can extend the result to \(\mathbb{P}^n\). Similar to how we worked with sets of three points, we can also find a projective transformation that maps \(n+1\) points in \(\mathbb{P}^n\) to another set of \(n+1\) points, providing that no sets of points lie on a hyperplane. To achieve this, we can follow similar steps as before, working with an \((n+1)\times(n+1)\) matrix instead of a \(3\times3\) matrix. Again, we can define the matrix by solving an analogous system of equations that relates the homogeneous coordinates of the points.

Key concepts

Projective Space

In geometry, projective space is a concept that extends the familiar notions of lines and planes to a more general framework. It is formed by adding 'points at infinity' to the common Euclidean spaces, allowing for a seamless handling of parallel lines intersecting at infinity and unifying certain geometric properties. Specifically, in a projective space, any two distinct lines in a plane intersect at a unique point, which can be a point at infinity if the lines are parallel in Euclidean space.

The exercise at hand deals with the two-dimensional projective space, usually denoted as \(\mathbb{P}^{2}\). It is the set of all lines through the origin in three-dimensional space, where each line corresponds to a point in \(\mathbb{P}^{2}\). This abstraction has profound implications in various applications ranging from computer graphics to complex algebraic geometry. A crucial element of working within this space is dealing with points and their coordinates in a way that reflects the new properties of this extended space, which brings us to the concept of homogeneous coordinates.

Homogeneous Coordinates

The concept of homogeneous coordinates is central to working in projective space. Unlike Cartesian coordinates, which represent a point in space with a pair (for two dimensions) or triplet (for three dimensions) of numbers, homogeneous coordinates introduce an additional dimension to account for the scale of the coordinates without changing the point's location in projective space.

For example, in the two-dimensional projective space \(\mathbb{P}^{2}\), a point is represented by a triplet \( [x : y : z] \) where \(x\), \(y\), and \(z\) are not all zero, and each triplet is considered equivalent to any other obtained by multiplying all three coordinates by a non-zero scalar. The 'colon' notation emphasizes that the coordinates are ratios, meaning \( [2 : 3 : 4] \) refers to the same projective point as \( [1 : 1.5 : 2] \) since each coordinate is doubled. When using homogeneous coordinates, we can easily represent points at infinity, which correspond to the coordinate \(z = 0\) in this case.

This representation is particularly robust when applying transformations, as it provides a unified approach for dealing with translations, rotations, scaling, and perspective projections.

Transformation Matrix

A transformation matrix is a powerful mathematical tool used to define and perform transformations on points within projective space. Essentially, it is a matrix that, when multiplied by the homogeneous coordinate vector of a point, gives the coordinates of the transformed point.

In our exercise, the transformation matrix, denoted as \(M\), is a non-singular \(3\times3\) matrix that represents a projective transformation in \(\mathbb{P}^{2}\). To maintain the properties of projective space after the transformation, the matrix must be invertible, which translates to the condition that it should not have a determinant of zero.

The challenge within the exercise is to construct such a matrix \(M\) that will map a set of three points \(P_{1}, P_{2}, P_{3}\) to another set \(Q_{1}, Q_{2}, Q_{3}\), based on their homogeneous coordinates. We achieve this by setting up a system of equations that characterize the image of each point under the transformation and solving for the entries of the matrix. With a transformation matrix, geometric transformations can be applied consistently and efficiently, making it a foundational element in many fields, including computer graphics, robotics, and photogrammetry.