Question 2.38: Determine all surfaces z = z(x,y) with the property that in ......
Determine all surfaces z = z(x,y) with the property that in every surface point M(x, y, z) the scalar product of the unit normal vector and the vector \vec{OM} is equal to
\frac{z-H(p, q)}{\left(1+p^2+q^2\right)^{1 / 2}}
where H is a homogeneous function of degree n ≠ 1, i.e.
H(t p, t q)=t^n H(p, q) \text { for all } t>0 \text {, }
and, moreover, it is assumed that the direction of the normal vector has been chosen so that it makes an acute angle with the positive direction of the x-axis.
If we denote by i, j and k, respectively, the unit vectors on the x-, y- and z-axis, then by assumption it holds
(x \mathbf{i} +y \mathbf{j} +z \mathbf{k} ) \cdot \frac{-p \mathbf{i}-q\mathbf{j} + \mathbf{k} }{\sqrt{1+p^2+q^2}}=\frac{z-H(p, q)}{\left(1+p^2+q^2\right)^{1 / 2}} .
Thus we get
px + qy = H(p, q).
Using the homogeneity of H, we obtain the complete solution
z=\frac{n-1}{n} \frac{(x+a y)^{n /(n-1)}}{(H(1, b))^{n /(n-1)}}+b
It is left to the reader to find the general solution; clearly, there is no singular solution.