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The length of the curve is given by the formula = | ′ | where | ′ | is the Euclidean norm of the tangent vector ′ to the curve. To justify this formula, define the arc length as limit of the sum of linear segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity.
The resulting length is shorter than the rest length, and is given by the formula for length contraction (with γ being the Lorentz factor): L = L 0 γ . {\displaystyle L={\frac {L_{0}}{\gamma }}.} In comparison, the invariant proper distance between two arbitrary events happening at the endpoints of the same object is given by:
This is the formula for length contraction. As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is ℓ. The proper length of an object is the length of the object in the frame in which the object is at rest.
The arc length, from the familiar geometry of a circle, is = The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of ):
The distance from (x 0, y 0) to this line is measured along a vertical line segment of length |y 0 - (-c/b)| = |by 0 + c| / |b| in accordance with the formula. Similarly, for vertical lines ( b = 0) the distance between the same point and the line is | ax 0 + c | / | a |, as measured along a horizontal line segment.
Analytically, the equation of a standard ellipse centered at the origin with width and height is: + = ... The length of the chord through one focus, ...
A more complicated formula, ... the Euclidean distance should be distinguished from the geodesic distance, the length of a shortest curve that belongs to the surface.
Using the arc length formula above, this equation can be rewritten in terms of dθ / dt : = =, =, where h is the vertical distance the pendulum fell. Look at Figure 2, which presents the trigonometry of a simple pendulum.