Space geometric transformation:
1. Homogeneous coordinates: an n-dimensional vector is represented by an n+1-dimensional vector. There are two main advantages of using homogeneous coordinates. One provides matrix operations to convert two-dimensional, three-dimensional and even high-dimensional An efficient way to transform a set of points in space from one coordinate system to another. Two can represent the point at infinity.
2. Projective transformation is the most generalized linear transformation. The projective transformation of n-dimensional projective space can be expressed algebraically as py = Tpx, where p is a scale factor, x and y are the homogeneous coordinates of the space points before and after the transformation, x =(x1,x2...xn+1)T,y is the same, Tp is a full rank (n+1)*(n+1) matrix,
3. Affine transformation, the projective center plane becomes At infinity, the projective transformation becomes an affine transformation.
4. The scale transformation is the Euclidean transformation with a scale factor
5. The Euclidean transformation is a transformation in the Euclidean space, which is very similar to the scale transformation, except that the scale factor is 1, and the Euclidean transformation has 6 degrees of freedom, of which 3 1 rotation, 3 translations.
Two Invariants of geometric transformation
1. Simple ratio: Three points A, B, C on the straight line L, with A and B as the basic points, C as the dividing point, two directed points determined by the dividing point and the basic point A line segment is simply called a simple ratio. Denoted as SR (A, B; C) = AC/BC
2. Intersection ratio: The ratio of two simple ratios in four points on a straight line is called the intersection ratio. CR(A,B;C,D) = SR(A,B;C)/SR(A,B;D) = AC/BC/(AD/BD)
3. Any 4 straight lines with point o as the intersection The cross ratio is called the line speed cross ratio CR(l1,l2;l3,l4) = sin(l1,l3)/sin(l2,l3)/(sin(l1,l4)/sin(l2,l4))
5 . Invariants
· Homotropism and junction are properties of projective invariance.
· Keep the intersection ratio of the point column on the line unchanged.
·Keep the cross ratio of the wiring harness unchanged.
1. Homogeneous coordinates: an n-dimensional vector is represented by an n+1-dimensional vector. There are two main advantages of using homogeneous coordinates. One provides matrix operations to convert two-dimensional, three-dimensional and even high-dimensional An efficient way to transform a set of points in space from one coordinate system to another. Two can represent the point at infinity.
2. Projective transformation is the most generalized linear transformation. The projective transformation of n-dimensional projective space can be expressed algebraically as py = Tpx, where p is a scale factor, x and y are the homogeneous coordinates of the space points before and after the transformation, x =(x1,x2...xn+1)T,y is the same, Tp is a full rank (n+1)*(n+1) matrix,
3. Affine transformation, the projective center plane becomes At infinity, the projective transformation becomes an affine transformation.
4. The scale transformation is the Euclidean transformation with a scale factor
5. The Euclidean transformation is a transformation in the Euclidean space, which is very similar to the scale transformation, except that the scale factor is 1, and the Euclidean transformation has 6 degrees of freedom, of which 3 1 rotation, 3 translations.
Two Invariants of geometric transformation
1. Simple ratio: Three points A, B, C on the straight line L, with A and B as the basic points, C as the dividing point, two directed points determined by the dividing point and the basic point A line segment is simply called a simple ratio. Denoted as SR (A, B; C) = AC/BC
2. Intersection ratio: The ratio of two simple ratios in four points on a straight line is called the intersection ratio. CR(A,B;C,D) = SR(A,B;C)/SR(A,B;D) = AC/BC/(AD/BD)
3. Any 4 straight lines with point o as the intersection The cross ratio is called the line speed cross ratio CR(l1,l2;l3,l4) = sin(l1,l3)/sin(l2,l3)/(sin(l1,l4)/sin(l2,l4))
5 . Invariants
· Homotropism and junction are properties of projective invariance.
· Keep the intersection ratio of the point column on the line unchanged.
·Keep the cross ratio of the wiring harness unchanged.
·If there are four straight lines of a bundle in the plane that are intercepted by any straight line, the intersection ratio of the intersection point column and the bundle intersection ratio are equal.
The point-column intersection ratio is the basic invariant of projective transformation and a necessary and sufficient condition for projective transformation, and the collinear four-point intersection ratio has the following characteristics:
1.CR(A,B;C,D) = CR(C,D;A,B);
2.CR(A,B;C,D) = CR(B,A;D,C)
3.CR(A,B;C,D) = 1-CR(A,B;C,D) = 1/CR(B,A;D,C)
4.CR(A,B;C,D) = 1-CR(A,C;B,D) = 1-CR(D,B;C,A)
Affine transformation
In addition to the invariance of the above projective transformation, the affine transformation has the following characteristics:
1. The parallelism between two straight lines is an affine invariant.
2. The simple ratio of three collinear points is the basic invariant of affine transformation.
3. The ratio of the areas of two triangles is affine invariant.
4. The ratio of the area enclosed by two closed curves is an affine invariant.
1. The parallelism between two straight lines is an affine invariant.
2. The simple ratio of three collinear points is the basic invariant of affine transformation.
3. The ratio of the areas of two triangles is affine invariant.
4. The ratio of the area enclosed by two closed curves is an affine invariant.