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1.2 Geometry in the Tangent Plane

In order to study the geometry of the tangent plane at a point P on a surface, we need a notion of the angle between two non-zero vectors V and W. This is obtained by using the first fundamental form on the surface to define
cos() = I(V,W)/(I(V)I(W)).
In particular, the angle i between a vector V and the vector Xi is given by cos(i) = VXi/(I( V)I(xi)) = ajgij/(giiI(V)) with no summation on the index i.
Two vectors V and W in the tangent plane at P are orthogonal if I(V,W) = 0. If V and W are orthogonal, so are the scalar multiples aV and bW. The totality of all non-zero scalar multiples of a vector V is called the direction determined by V, so a direction is determined by the ratio a1:a2 of its components. Orthogonality is therefore a property of a pair of directions.
Suppose that two directions are determined by the quadratic condition cijaiaj = 0 where cij = cji. Then these directions are orthogonal if and only if g22c11 - 2g12c12 + g11c22 = 0. Then the quadratic condition is c11 + 2c12x + c22x2 = 0, so the solutions p and q satisfy pq = c11/c22 and p+q = -2c12/c22. Let V = aiXi and W = bjXj be two non-zero vectors with directions and set p = a2/a1 and q = b2/b1. Then I(V,W)/(a1b1) = g11 + (b2/b1)g12 + (a2/a1)g21 + (a2/a1)(b2/b1)g22 = g11 + (u + v)g12 + uvg22 = g11 +(-2c12/c22) g12 + c11g22 = (1/c22) (g22c11 - 2g12c12 + g11c22), so I(V,W) = 0 if and only if g22c11 - 2g12c12 + g11c22 = 0.

Exercises for 1.2

  1. The curves defined by ui = constant for a fixed index i are called parametric curves of the surface. There are therefore two families of parametric curves on a surface, and through each point of the surface there passes exactly one parametric curve from each family. Find the condition that the parametric curves are orthogonal. Under which conditions will the parametric curves meet at a 45° angle?
  1. For each of the surfaces in the problem set for section 1.1, find the angle between parametric curves.
  1. Consider a family of curves given by the conditions ciai = 0, for a fixed set of constants ci. Show that the orthogonal trajectories of this family satisfy the condition (g11c1 - g12c2)a1 + (g12c1 - g22c2)a2 = 0. Use this formula to find the orthogonal trajectories of the family of circles given in polar coordinates by r = c cos(). Sketch the curves in this family and sketch their orthogonal trajectories.
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