What are homogeneous coordinates and why are they used?
Homogeneous coordinates are an extended way to represent points (and lines or planes) using one extra coordinate. For example, a 2D euclidean coordinate is (x,y) while a 2D homogeneous coordinate is (x,y,1).
They are used because they let us describe geometry using simple matrix multiplication and therefore make calculations in Computer VIsion much more easy. Homogeneous coordinates make transformations linear, which means you can apply transformations with matrix multiplication (-> linear algebra can be used).
Given are two points x1 and x2 in homogeneous coordinates. Which line connects both points?
x1 = (2, -3, 1)^T and x2 = (0, -4, 2)^T
l = (-2, -4, -8)
Calculate the cross product between both points to get the answer.
What are ideal points and ideal lines?
Ideal points are vanishing points/points at infinity with (u, v, 0). They can’t be expressed in euclidean cordinates since x = u/w and y = v/w (which is not possible if w = 0).
The intersection of two parallel lines is an ideal point/vanishing point.
Ideal lines are vanishing lines/lines at infinity with (0, 0, c).
If x is an non-ideal point, Hx is also a non-ideal point. True or false?
False. Why?
Every homogeneous line contains at least one ideal point. True or false?
True.
A point (u, v, w) is on the line (a, b , c) if au + bv + cw = 0.
An ideal point has the coordinates (u, v, 0).
Therefore the equation au + bv + cw = 0 must satisfy au + bv = 0 for the statement to be true. Since the coordinates of the ideal point are not specified, they could for example be (b, -a, 0) which would make au + bv = 0.
Some non-ideal points lie on an ideal line. True or False?
False.
An non-ideal point has the coordinates (u, v, w) with w != 0.
An ideal line has the coordinates (0, 0, c).
Therefore: au + bv + cw = 0 = 0u + 0v + cw. Since c and w cannot be 0, there is no way a non-ideal point can be on an ideal line.
The line that goes through two ideal points is always an ideal line. True or false?
The line that goes through two points is calculated with the cross product between the two points.
An ideal point is expressed as (u, v, 0).
The cross product between two vectors (a1, a2, a3) and (b1, b2, b3) is expressed as:
(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).
When both points are ideal points, a3 = b3 = 0.
Then:
(a20 - a0b2, a0b1 - a10, a1b2 - a2b1) = (0, 0, a1b2 - a2b1) which represents an ideal line, since an ideal line is expressed as (0, 0, c).
The intersection between an ideal line and a non-ideal line is always an ideal point. True or False?
True!
The intersection between an ideal line (0, 0, a3) and a non-ideal line (b1, b2, b3) is calculated using the cross product x = l1 x l2.
The last line of the cross product is a1b2 - a2b1. Since a1 = a2 = 0, this means w = 0, which makes the intersection point an ideal point.
Any two homogeneous lines intersect in an ideal point. True or False?
False, since the intersection of two lines is calculated using the cross product between them, and for an ideal point, w must be 0.
But w is calculated using a1b2 - a2b1, which can or cannot be 0, depending on the lines.
given is a line (…,.). Give the implicit as well as the intersect and .. form.
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