Pre- and co-image (Lecture)
Lecture Notes
See also Notes from 2021.
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
- functions \[\ell = \{ \vec{x}=(x,y) | y = a\cdot x, x\in\mathbb{R} \}\] for some \(a\in\mathbb{R}\)
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 \(\ell^\bot\)
- The plane needs one equation \[\mathcal{P}=\{\vec{x} | \vec{x}\cdot\vec{x}^\bot=0 \}\]
- 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
- one function per dimension; or
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 \(l\)