Blaga P. Lectures on the differential geometry of - tiera.ru

220 Problems
23. Find the tangent plane to the surface z = x 4 − 2xy 3 , perpendicular to the vector
a = {−2, 6, 1}. Find, also, the tangency point.
24. Find the envelope and the regression edge of the family of spheres of constant radius,
equal to a, with the centers on the circle
⎧
⎪⎨ x
⎪⎩
2 + y2 = b2 ,
z = 0.
25. Find the envelope of the family of spheres
x 2 + y 2 + (z − C) 2 = 1.
26. Find the envelope, the characteristics and the regression edge for a family of spheres
of constant radius a, with centers on a given curve ρ = ρ(s) (tubular surface).
27. Find the envelopes and the regression edge for a family of spheres passing through
the origin of the coordinates and with the centers on the curve
⎧
x = t
⎪⎨
⎪⎩
3 ,
y = t2 ,
z = t.
28. Find the envelope of a family of planes forming with the coordinate solid angle
x > 0, y > 0, z > 0 a triangular pyramid of constant volume V.
29. Find the envelope and the regression edge for the family of normal planes to the
helix ⎧⎪⎨⎪⎩
x = a cos t,
y = a sin t,
z = bt.
30. Find the envelopes, the characteristics and the regression edge for the family of
normal planes to a space curve ρ = ρ(s) (the polar surface of the curve).
31. Find the envelope, the characteristics and the regression edge for the family of rectifying
planes of a space curve ρ = ρ(s).
32. Find the envelope, the characteristics and the regression edge of the family of osculating
planes of a space curve ρ = ρ(s).