Straight Line Segment is Shortest Path between Two Points

== Theorem ==
Let $A$ and $B$ be points in a Euclidean space.
Let $\LL$ be the straight line segment lying on both $A$ and $B$.
Then $\LL$ is the shortest line that lies on both $A$ and $B$.
== Proof ==
Let $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ be an arbitrary pair of points embedded in the Cartesian $\R^2$-plane.
Let $\LL$ be the straight line segment from $A$ to $B$.
We are to demonstrate that there does not exist a differentiable real function $f: \R \to \R$ such that:
and such that the arc length of the graph of $f$ from $A$ to $B$ is less than the length of $\LL$.
Note that for $x_1 = x_2$, a similar argument can applied to a function $\map f y$.
The proof relies on the Fundamental Theorem of Calculus and the definition of the the arc length of a differentiable real function.
 there exists a real function $f: \R \to \R$ such that:
:$\map f {x_1} = y_1$ and $\map f {x_2} = y_2$
and such that:
:the arc length of the graph of $f$ from $A$ to $B$ is less than the length of $\LL$.
This means that $\map f x$ is continuous over $\closedint {x_1} {x_2}$ (and for the sake of the proof, differentiable over the same interval).
Thus, the arc length of the graph of $\map f x$ over $\closedint {x_1} {x_2}$ is:
:$\ds L_f = \int_{x_1}^{x_2} \sqrt {1 + \paren {\map {f'} x}^2} \rd x$
Suppose that $L_f$ is less than the length of the line connecting $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.
We have that Slope of Tangent to Curve at Point equals Value of Derivative, which we will denote $s$.
Thus, the length of the line is given by:
:$L_l = \ds \int_{x_1}^{x_2} \sqrt {1 + s^2} \rd x$
where $s = \dfrac {y_2 - y_1} {x_2 - x_1}$.
Since $L_f < L_l$, transitively, we have that:
:$\ds \int_{x_1}^{x_2} \sqrt {1 + \paren {\map {f'} x}^2} \rd x < \int_{x_1}^{x_2} \sqrt {1 + s^2} \rd x$
=== Lemma $1$ ===
=== Lemma $2$ ===
Following from and , we can reduce the inequality into:
:$\ds \int_{x_1}^{x_2} \size {\map {f'} x} \rd x < \int_{x_1}^{x_2} \size s \rd x$
Since $s$ is constant, we can reduce the using the Fundamental Theorem of Calculus to:
:$\ds \int_{x_1}^{x_2} \size {\map {f'} x} \rd x < \size {y_2 - y_1}$
There are now four possibilities:
:$(1): \quad \map {f'} x$ is strictly positive over $\closedint {x_1} {x_2}$
:$(2): \quad \map {f'} x$ is strictly negative over $\closedint {x_1} {x_2}$
:$(3): \quad \map {f'} x$ is both negative and positive over $\closedint {x_1} {x_2}$
:$(4): \quad \map {f'} x$ is $0$ over $\closedint {x_1} {x_2}$
The following argument applies to both Case $1$ and Case $2$ without loss of generality, and Case $3$ is dealt with similarly.
=== Case 1 ===
For Case $1$, we have since $\size a = a$ for positive, real $a$:
:$\ds \int_{x_1}^{x_2} \map {f'} x \rd x < y_2 - y_1$
Using the Fundamental Theorem of Calculus we have that:
:$\map f {x_2} - \map f {x_1} < y_2 - y_1$
By our previous assumptions that $\map f x$ connects $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$:
:$y_2 - y_1 < y_2 - y_1$
which is a contradiction.
=== Case 2 ===
For Case $2$, we have that $s$ must be negative.
Otherwise $\map f x$ is strictly decreasing over $\closedint {x_1} {x_2}$ but $y_2 > y_1$.
This would immediately contradict that $\map f {x_1} = y_1$ or $\map f {x_2} = y_2$.
Thus, we can flip both signs and apply the reasoning for Case $1$.
=== Case 3 ===
With Case $3$, we have that since $\size a \ge a$ for all real $a$:
:$\ds \int_a^b \size {\map f x} \rd x \ge \int_a^b \map f x \rd x$
From this the proof immediately follows from Cases $1$ and $2$.
=== Case 4 ===
With Case $4$, the argument is applied using Case $1$, since $\size 0 = 0$.
=== When $x_1 = x_2$ ===
When $x_1 = x_2$, we will take a function $f(y)$ to measure the arc-length instead of $f(x)$. From this alteration, which is distance preserving, we can apply our previous reasoning. It is distance preserving since we are taking the sum of $\sqrt{\d x^2 + \d y^2}$ instead of $\sqrt{\d y^2 + \d x^2}$, which are equivalent.
All cases have been covered, and the result follows.
Category:Straight Line Segment is Shortest Path between Two Points
Category:Euclidean Geometry
Category:Straight Lines