1, the definition and representation of space vector
Space vector is a vector in three-dimensional space, which can be expressed as an ordered triple composed of three components. The common representation method is to use letters and arrows (such as AB→) to represent the vector, and the starting point and ending point of the vector respectively indicate the position of the vector in space.
2. The concept of distance from point to surface
The distance from point to surface refers to the shortest distance from a point to a plane in space. This distance can be calculated by vector operation and geometric derivation.
3. Derivation of point-to-surface distance formula
In order to calculate the distance from a point to a surface, we can derive the corresponding formula by using the properties of vectors and plane equations. Among them, the key idea is to transform the distance from point to plane into the vertical distance from point to point on the plane.
4. Distance formula from point to surface
Through vector operation and geometric derivation, the distance formula from point to surface can be obtained: the distance formula from point P to plane Ax+By+Cz+D=0 is: d=|Ax? +By？ +Cz？ +D|/√(A？ +B？ +C？ ) among them, (x? ，y？ ，z？ ) is the coordinate of any point on the plane, and a, b, c and d are the parameters of the plane.
5. The application of point-to-surface distance formula
The distance formula from point to surface is widely used in computer graphics, machine learning, physics and other fields. In computer graphics, collision detection, projection transformation and other operations can be carried out through the distance formula from point to surface. In machine learning, the distance from point to surface can be used as one of the characteristics of classification and regression tasks. In physics, this formula can be used to calculate the distance and collision between particles and a plane.
6. Derivation process of point-to-surface distance formula
The derivation process of point-to-surface distance formula is based on the properties and geometric derivation of vectors. The key idea is to find a vector perpendicular to the plane and passing through the point to be solved, and then calculate the projection length of the vector on the plane normal vector.
To sum up, the distance formula of point-to-surface space vector is a mathematical formula derived from vector operation and geometry. This formula can help us calculate the shortest distance from a point to a plane, and it is widely used in spatial analysis and geometric calculation.