Monotonicity of the Line Function K on a 3D Space

[Asuralm]

Hi all:

Assume in 3D space there is a point [itex]v=[v_x, v_y, v_z][/tex], and a normal vector associate with it as $$n=[n_x, n_y, n_z]$$. A line function is defined as $$u=w+t\cdot l$$ where $$w=[w_x, w_y, w_z]$$ is a point, and $$l=[l_x, l_y, l_z]$$ is the normalized direction of the line. l and n are normalized. Assume there is a function defined as:<br /> <br /> $$K = \frac{(u-v)\cdot n}{||u-v||^2}$$<br /> <br /> My question is when point u varies on the line, is the function K varies monotonically?<br /> <br /> I've tried to compute $$\frac{dK}{dt}$$, but I can't really see if it's monotone or not, can some one help me please?<br /> <br /> Thanks[/itex]

 

[tiny-tim]

Hi Asuralm!

Forget calculus, this is geometry …

Hunt: if u and v represent points U and V, and if the nearest point on L to U is N, what is (u - v).n ?