Focal Points

• J. A. Thorpe
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

The construction in the proof of Theorem 2 of the previous chapter surrounds the parametrized n-surface $$\varphi :\,U\, \to \,{\mathbb{R}^{n + 1}}$$ with a family of smooth maps $${\varphi _s}:\,U\, \to \,{\mathbb{R}^{n + 1}}\left( {s\, \in \,\mathbb{R}} \right)$$ given by
$${\varphi _s}\left( q \right)\, = \,\psi \left( {q,\,s} \right)\, = \,\varphi \left( q \right)\, + \,s{N^\varphi }\left( q \right)$$
(Figure 15-6). When s = 0, ϕ s = ϕ is a parametrized n-surface in ℝ n +1. For s ≠ 0. however, ϕ s may fail to be a parametrized n-surface because there may be points pU at which ϕ s fails to be regular. At each such point there will be a direction $$v\, \in \,\mathbb{R}_p^n\,\left( {\left\| v \right\|\, = \,1} \right)$$ such that $$d{\varphi _s}\left( v \right)\, = \,0$$. If α is a parametrized curve in U with $$\dot \alpha \left( {{t_0}} \right)\, = \,v$$, it follows that
$${\varphi _s}\,\dot \circ \,\alpha \left( {{t_0}} \right)\, = \,d{\varphi _s}\left( {\dot \alpha \left( {{t_0}} \right)} \right)\, = \,0;$$
that is, the curve $${\varphi _s}\, \circ \,\alpha \left( t \right)\, = \,\varphi \left( {\alpha \left( t \right)} \right)\, + \,s{N^\varphi }\left( {\alpha \left( t \right)} \right)$$ pauses (has velocity zero) at t = t0. Geometrically, this says that the normal lines which start along the curve ϕ ∘ α near ϕ(p) = ϕ(α(t0)) tend to focus at f = ϕ s (α(t0)) = ϕ s (p) (see Figure 16.1). Such points f are called focal points of ϕ. Note that the normal lines along α need not actually meet at a focal point.