# Monotone Test of the equation

1. May 15, 2009

### 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:

$$K = \frac{(u-v)\cdot n}{||u-v||^2}$$

My question is when point u varies on the line, is the function K varies monotonically???

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?

Thanks

Last edited: May 15, 2009
2. May 15, 2009

### 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 ?