Collinearity equation

The collinearity equations are a set of two equations, used in photogrammetry and remote sensing to relate coordinates in a sensor plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from the central projection of a point of the object through the optical centre of the camera to the image on the sensor plane. The collinearity equations are a set of two equations, used in photogrammetry and remote sensing to relate coordinates in a sensor plane (in two dimensions) to object coordinates (in three dimensions). The equations originate from the central projection of a point of the object through the optical centre of the camera to the image on the sensor plane. Let x,y, and z refer to a coordinate system with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by x P , y P , z P {displaystyle x_{P},y_{P},z_{P}} , the coordinates of the image point of P on the sensor plane by x and y and the coordinates of the projection (optical) centre by x 0 , y 0 , z 0 {displaystyle x_{0},y_{0},z_{0}} . As a consequence of the projection method there is the same fixed ratio λ {displaystyle lambda } between x − x 0 {displaystyle x-x_{0}} and x 0 − x P {displaystyle x_{0}-x_{P}} , y − y 0 {displaystyle y-y_{0}} and y 0 − y P {displaystyle y_{0}-y_{P}} , and the distance of the projection centre to the sensor plane z 0 = c {displaystyle z_{0}=c} and z P − z 0 {displaystyle z_{P}-z_{0}} . Hence: Solving for λ {displaystyle lambda } in the last equation and entering it in the others yields: The point P is normally given in some coordinate system 'outside' the camera by the coordinates X, Y and Z, and the projection centre by X 0 , Y 0 , Z 0 {displaystyle X_{0},Y_{0},Z_{0}} . These coordinates may be transformed through a rotation and a translation to the system on the camera. The translation doesn't influence the differences of the coordinates, and the rotation, often called camera transform, is given by a 3×3-matrix R, transforming ( X − X 0 , Y − Y 0 , Z − Z 0 ) {displaystyle (X-X_{0},Y-Y_{0},Z-Z_{0})} into: