Spherical Coordinate (球面坐标). Since light are mostly expressed in terms ofdirections, it is generally more convenient todescribe them by spherical coordinates rather thanby cartesian coordinate vectors.As illustrated in the figure, a vector in sphericalcoordinates is specified by three elements.magnituder denotesthelength ofthevector.- measures the angle between the vector and the z-axis,- represents the counterclockwise angle on the x-y planefrom the X-axis to the projection of the vector onto the X-yplane
Spherical Coordinate (球面坐标) • Since light are mostly expressed in terms of directions, it is generally more convenient to describe them by spherical coordinates rather than by cartesian coordinate vectors. • As illustrated in the figure, a vector in spherical coordinates is specified by three elements. – magnitude r denotes the length of the vector. – Θ measures the angle between the vector and the z-axis, – ψ represents the counterclockwise angle on the x-y plane from the x-axis to the projection of the vector onto the xy plane
Spherical Coordinate (球面坐标)·Relationship between Cartesian(笛卡尔)and spherical coordinates- (x,y,z) <> (r, 0, y)A· Conversion= sqrt(x^2+y^2+z^2)0 = acos(z/r); = atan2(y,x);0: z=r cos(o); y = r sin(O)sin(y);: x = r sin(o)cos(y);dx
Spherical Coordinate (球面坐标) • Relationship between Cartesian(笛卡尔) and spherical coordinates – (x,y,z) (r, Θ, ψ) • Conversion • r = sqrt(x^2+y^2+z^2) • Θ = acos(z/r); • ψ = atan2(y,x); • z = r cos(Θ); • y = r sin(Θ)sin(ψ); • x = r sin(Θ)cos(ψ);
Solid Angle(立体角)Light generally arrives at or leaves a surfacepoint from a range of directions that isdenoted by solid angles. solid anglesrepresents a 3D generalization of angleformed by a region on a sphere.dsdo=2. Max value of arsinesolid angle is 4元which is given bya sphere.(b)(a)
Solid Angle(立体角) • Light generally arrives at or leaves a surface point from a range of directions that is denoted by solid angles. solid angles represents a 3D generalization of angle formed by a region on a sphere. • Max value of a solid angle is , which is given by a sphere. 4 2 ds d r
Solid Angle(立体角): For a differential solid angle described bydifferential angles do,dp in the ,Φdirections, its differential area dA on thesphere isdA =(rdO)(r sinOdp) = r? sinOd0dp: From the solid angle definition, thedifferential solid angle is given by:dAdosinddp
Solid Angle(立体角) • For a differential solid angle described by differential angles in the directions, its differential area dA on the sphere is • From the solid angle definition, the differential solid angle is given by: 2 sin dA d d d r 2 dA rd r d r d d ( )( sin ) sin d,d ,
Foreshortened Area（投影面积）: The apparent area of a surface patch accordingto the angle at which it is viewed For a surface patch of area A, its foreshortenedarea from direction Ois given as A cos(), sinceits apparent length in the x direction is scaledby cos(0).nA,coseAForeshortenedArea = Acos 0area=AcosoAArea=A=AxA
Foreshortened Area（投影面积） • The apparent area of a surface patch according to the angle at which it is viewed • For a surface patch of area A, its foreshortened area from direction θis given as A cos(θ), since its apparent length in the x direction is scaled by cos(θ). Area A cos