0

Based on this answer for the previous question I had asked on august of last year, the first difficulty for me to understand is why it is a condition that the circle must 'touch' the other curve?

Leaving the above point aside, it is clear to me that if we have two curves $C_1$ and $C_2$ , we can use the circle argument for a point on $C_2$ to find that nearest point's line segment is along the normal direction to curve $C_1$ and vice versa for $C_2$, but how is it possible that we can show that for the 'shortest distance' between two curves it must be that it is along mutual normal of two curves?

As in, we know shortest distance from one curve to a point is along it's normal, but , how can we guarantee that shortest distance between two curves is along mutual normal?

tryst with freedom
  • 11,538
  • Let $A_1 \in C_1$ and $A_2 \in C_2$ the points attaining the minimum distance between the two curves. Then the segment $A_1 A_2$ is the distance between $A_1$ and $C_2$, hence it is normal to $C_2$. The same for the other curve. – Crostul Feb 03 '21 at 18:04

1 Answers1

1

Look at this figure and just think about it:

enter image description here

David G. Stork
  • 29,774