- denote 表示
- skew-symmetric 反对称的
- arithmetic 算术的
- polynomial 多项式的
- quadratic 二次的
- conic 圆锥的
- augmented vector 扩展矢量
- ellipsoid 椭圆体
- cylinder 圆柱体
2D Points
- 2D points can be inhomogeneous coordinates as
- or in homogeneous coordinates as
- where is called projective space.
- Remark: Homogeneous vectors that differ only by scale are considered equivalent and define an equivalence class. Homogeneous vectors are defined only up to scale
2D Lines
- 2D Lines can also be expressed using homogeneous coordinates
- We can normalize so that with . In this case, is the normal vector perpendicular to the line and is its distance to the origin.
- An exception is the line at infinity which passes through all ideal points.
Cross Product
- Cross product expressed as the product of a skew-symmetric matrix and a vector:
- Remark: In this course, we use squared brackets to distinguish matrices from vectors.
2D Line Arithmetic
- In homogeneous coordinates, the intersection of two lines is given by
- Similarly, the line joining two points can be compactly written as;
- The symbol denotes the cross product. Proof is in the exercise:
3D Points
- 3D points can be written in inhomogeneous coordinates as
- or in homogeneous coordinates as
- with projective space
3D planes
- 3D planes can also be represented as homogeneous coordinates :
- Again, we can normalize so that with . In this case, is the normal perpendicular to the plane and is its distance to the origin.
- An exception is the plane at infinity which passes through all ideal points (= points at infinity) for which
3D Line
- 3D Lines are less elegant than either 2D lines or 3D planes. One possible representation is to express points on a line as a linear combination of two points and on the line:
- However, this representation uses 6 parameters for 4 degrees of freedom. Alternative minimal representations are the two-plane parameterization or Plücker coordinates. See Szeliski, Chapter 2.1 and Hartley/Zisserman, Chapter 2 for details.
3D Quadrics
- The 3D analog of 2D conics is a quadric surface:
- Useful in the study of multi-view geometry. Also serves as useful modeling primitives(spheres, ellipsoids, cylinders), see Hartley and Zisserman, Chapter 2 for details.
2D Transformations
- Translation(平移): (2D Translation of the input, 2 DoF(Degrees of Freedom))
- Using homogeneous representations allows to chain/invert transformations
- Augmented vectors can always be replaced by general homogeneous ones
- Euclidean(欧几里得): (2D Translation + 2D Rotation, 3 DoF)
- is an orthonormal(标准正交的) rotation matrix with and
- Euclidean transformations preserve Euclidean distances