16.3. ARC LENGTH AND ORIENTATIONS 337

This reduces to(r (t)−P 0) · (−r′ (t))

|r (t)−P 0|=

(r (t)−P 1) · (r′ (t))|r (t)−P 1|

(16.5)

Now(r (t)−P 1) · (r′ (t))

|r (t)−P 1|=

ddt

|r (t)−P 1|

and a similar formula holds for P 1 replaced with P 0. This is because

|r (t)−P 1|=√

(r (t)−P 1) · (r (t)−P 1)

and so using the chain rule and product rule,

ddt

|r (t)−P 1| =12((r (t)−P 1) · (r (t)−P 1))

−1/2 2((r (t)−P 1) ·r′ (t)

)=

(r (t)−P 1) · (r′ (t))|r (t)−P 1|

.

Therefore, from 16.5,

ddt

(|r (t)−P 1|)+ddt

(|r (t)−P 0|) = 0

showing that |r (t)−P 1|+ |r (t)−P 0|=C for some constant C.This implies the curve ofintersection of the plane with the room is an ellipse having P 0 and P 1 as the foci.

16.2.3 Leibniz’s Notation

Leibniz’s notation also generalizes routinely. For example, dydt = y′ (t) with other similar

notations holding.

16.3 Arc Length and OrientationsThe application of the integral considered here is the concept of the length of a curve.

Definition 16.3.1 C is a smooth curve in Rn if there exists an interval [a,b] ⊆ Rand functions xi : [a,b]→ R such that the following conditions hold

1. xi is continuous on [a,b].

2. x′i exists and is continuous and bounded on [a,b], with x′i (a) defined as the derivativefrom the right,

limh→0+

xi (a+h)− xi (a)h

,

and x′i (b) defined similarly as the derivative from the left.

3. For p(t)≡ (x1 (t) , · · · ,xn (t)), t → p(t) is one to one on (a,b).

4. |p′ (t)| ≡(

∑ni=1 |x′i (t)|

2)1/2

̸= 0 for all t ∈ [a,b].

5. C = ∪{(x1 (t) , · · · ,xn (t)) : t ∈ [a,b]}.