# Lecture Notes

## Linear objects in 2D

• The most important linear object is the line through the origin.
• These are subspaces of dimension one.
• The object is a set $$\ell\subset\mathbb{R}^2$$
• Three descriptions
• functions $\ell = \{ \vec{x}=(x,y) | y = a\cdot x, x\in\mathbb{R} \}$ for some $$a\in\mathbb{R}$$
• Exception: The vertical line would have $$a=\infty$$, for infinitely steep
• equations $\ell = \{ \vec{x}=(x,y) | \vec{x}\cdot\vec{x}^\bot \}$ for some $$\vec{x}^\bot\in\mathbb{R}^2$$
• Note that for $$c\neq0$$, $$\vec{x}^\bot$$ and $$c\vec{x}^\bot$$ define the same line.
• span $\ell = \{ \vec{x}=(x,y) | a\cdot \vec{x}_0, a\in\mathbb{R} \}$ for some $$\vec{x}_0\in\mathbb{R}^2$$
• Exception: The vertical line would have $$a=\infty$$, for infinitely steep

If we normalise $$\vec{x}^\bot$$, we can write $$\vec{x}^\bot=(a,1)$$ for $$a\in\mathbb{R}$$ unless we describe the vertical line, which has $$\vec{x}^\bot=(1,0)$$, which we could imagine writing $$(\infty,1)$$.

• We can normalise $$\vec{x}_0$$ in the same way.
• The set of lines through origo is equivalent to $$\mathbb{R}\cup\{\infty\}$$, which can be seen in either representation.

## Linear objects in 3D

We have the same situation in 3D, but we have more objects of interest.

• In 2D, the line is defined by one function or one equation.
• In 3D we have
• the line $$\ell= \{(x,y,z) | z = ax + by, (x,y)\in\mathbb{R}\}$$
• the plane $$\mathcal{P}= \{(x,y,z) | z = ax, y = bx, x\in\mathbb{R}^2\}$$ (two function)
• Using equations to define it
• The plane needs one equation $\mathcal{P}=\{\vec{x} | \vec{x}\cdot\vec{x}^\bot=0 \}$
• $$\vec{x}^\bot$$ is the dual space $$\mathcal{P}$$
• The line needs two equation $\ell=\{\vec{x} | \vec{x}\cdot\vec{y}_1=0, \vec{x}\cdot\vec{y}_1=0\}$
• The space spanned by $$\vec{y}_1$$ and $$\vec{y}_2$$ is the dual space $$\ell^\bot$$
• What does it look like as spans?
• An object needs
• one function per dimension; or
• Each adds one degree of freedom
• one equation per codimension
• Each equation removes one degree of freedom

## Projections from 3D to 2D

• Recall that each point $$x$$ in the image plane is the image of any point on a line through $$O$$
• Correspondence between lines through $$O$$ and point in the image.
• This line is called the pre-image of $$x$$.

Draw frontal model with image at $$Z=1$$. This gives projective image co-ordinage $$(x,y,1)$$ embedded in 3D.

• What about a line $$l$$ in the image plane? What is the pre-image?
• Plane $$P$$ through the origin. The line $$l$$ is the intersection of $$P$$ and the image plane
• What is the image of a line $$L$$ in 3D?
• if $$O\in L$$ we have a point, whose pre-image is $$L$$
• if $$P\not\in L$$, we have a line $$l$$ whose pre-image is a plane $$P\ni O$$
• $$P$$ is described by an orthogonal vector, the dual space $$P^\bot$$,
which we call the co-image of $$l$$

# Notes from the text book

The following notes were made 2021 based on a textbook. This exposition is not recommended because it is driven by the definitions which only gain meaning later in the course.

## Image and Image Plane

• Image Plane is the universe where the image lives

$\text{image}\subset\text{image plane}$

• The Image Plane is a 2D World
• The Image Plane exists in a 3D World

## Pre-image

• Preimage is the set of points in 3D projecting onto the Image Plane
• What is projection?
• draw a line through the 3D point and origo (the pinhole)
• the projection is the intersection with the image plane.
• Thus
• $$\text{preimage} = \mathsf{span}(\text{image})$$
• $$\text{image} = \text{preimage}\cap\text{image plane}$$
• The span of a set of points is the smallest linear subspace containing all the points

## Points and Lines

Image object Pre-Image
Point (dimension 0) Line through origo (dimension 1)
Line (dimension 1) Plane through origo (dimension 2)
• Preimage is a linear subspaces, i.e. includes origo
• A single point projects onto a point
• any other point on the same line through origo projects onto the same point
• A line projects onto a line if it does not pass through origo

## Co-image

• Coimage is the set of points (space) orthogonal on the preimage

$\text{coimage} = \text{preimage}^\bot$

$\text{preimage} = \text{coimage}^\bot$

## Points and Lines

Image object Pre-Image Co-Image
Point (dimension 0) Line through origo (dimension 1) Plane (co-dimension 1)
Line (dimension 1) Plane through origo (dimension 2) Line (co-dimension 2)
• Preimage and coimage are linear subspaces
• origo is in both the pre- and co-image

# Notation

• Recall $$\hat u$$ is a skew-symmetric matrix associated with $$u$$
• $$\mathsf{span}(\hat u) = u^\bot$$
• Associate an image point $$x$$ with either its pre-image or co-image

# Systems of Equations and Orthogonal Vectors

• $$\ell^Tx=0$$ is an equation in three unknowns
• This defines a plane (two unknowns)
• e.g. $$x_1+ax_2+bx_3$$
• If you have two points, say, $$\ell^TL=0$$, you have two equations
• This defines a line (one unknowns)
• e.g. $$x_1+ax_2$$
• If you have two points $$x_1$$ and $$x_2$$ on a line
• $$x_1\times x_2$$ is orthogonal on both of them
• and on any other point on the line