Jump measurement optimizations

We stand on the shoulders of those who have come before us and created jump measurement procedures. Ullr has built on their work by finding an opportunity to reduce the number of measurements taken in the field while still delivering all of the same data.

In studying the current jump measurement procedure, we found that Takeoff Length, Takeoff Height, and Takeoff Angle measurements can all be derived from Takeoff Transition measurements.

In the Ullr app, we do the math for you. The goal of this blog post is to prove that the simplified Ullr process is equivalent to the standard process with both a mathematical proof and an intuitive explanation.

The intuitive explanation

If we draw a jump's transition like this:

[image] - bare transition profile

And we measure slope of the transition in two foot increments, then we can draw those slopes like this:

[image] - transition profile with two foot line segments

From here, imagine that each of these slopes is actually the hypotenuse of a right triangle. For each of thee triangles, because we know the length of the hypotenuse and it's slope, we can use some trigonometry to calculate the length of the other two sides of the triangle.

[image] - transition profile with two foot line segments completed as triangles. Trig equations pointing to each rise and run

Now we know the length of every side of every triangle. Imagine now that we move each of the vertical segments all the way to the right, and each of the horizontal triangle segments all the way to the bottom. Now we can add all of the triangle sides together to get the total horizontal length of the takeoff and the total vertical rise.

[image] - transition profile with line segments moved out for total rise and run

Remember learning about the Pythagorean Theorem in school and wondering when you'd ever use it in real life? Well, this is your big moment, baby! We can use the Pythagorean Theorem (a² + b² = c²) to find the Takeoff Length. We can also use some trigonometry to find the angle of Takeoff tan(theta) = opposite / adjacent

[image] - transition profile with takeoff length and angle calculated

That leaves us with only the takeoff angle left to calculate. BUT, we've already taken this measurement as part of the transition slope measurements (it should be the first angle measurement that you took)

The mathematic explanation

Let's start by reducing the shape of a jump's takeoff to a curve in 2D space where the curve represents the profile of the takeoff. Let's also list some properties about the takeoff profile curve that we can take as givens:

1. Starting and Ending Slopes:

   • The left (starting) end of the curve has a tangent that makes an angle of 0° (i.e., horizontal).

   • The right (ending) end of the curve has a tangent that makes an angle of at most 90° (i.e., non‐vertical).

2. Smoothly Increasing Slope: The slope (or, equivalently, the tangent angle) increases continuously and "smoothly" from 0° at the left end to its final value on the right.

3. Discrete Slope Data: The slope (in degrees) is given (or measured) at every 2‑foot horizontal interval along the curve.

1. Setting Up the Problem

Consider a curve represented by a function:

y = f(x), for x ∈ [0, L]

with the following conditions:

• Initial Position: We set f(0) = 0 so that the left end is at the origin.
• Initial Slope: The curve starts with a horizontal tangent, i.e., f'(0) = 0 (0° slope).
• Smooth Increase in Slope: The derivative f'(x) is monotonically non‑decreasing over [0, L]. This condition ensures the curve is convex.

At the right end of the interval, the tangent makes an angle θ_L with the horizontal, where:

0° ≤ θ_L ≤ 90°

and the derivative at x = L is given by:

f'(L) = tan(θ_L)

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2. Enclosing the Curve in a Right Triangle

Constructing the Triangle

We build a right triangle in the xy–plane with:

• Horizontal Leg (Base): The line segment from (0, 0) to (L, 0) with length L.
• Vertical Leg (Height): The line segment from (L, 0) to (L, f(L)) with length f(L).
• Hypotenuse: The line segment connecting (0, 0) and (L, f(L)).

The hypotenuse has a length:

h = √(L² + f(L)²)

and it makes an angle α with the horizontal:

α = arctan(f(L) / L)

Demonstrating That the Curve Lies Within the Triangle

Because f(x) is convex and starts at (0,0), a fundamental property of convex functions tells us that:

f(x) ≤ (x/L) · f(L) for every x ∈ [0, L]

Proof of the Inequality:

• Let x = tL for t ∈ [0,1].
• By convexity: f(tL) ≤ (1-t)f(0) + t·f(L)
• Since f(0) = 0, it simplifies to: f(tL) ≤ t·f(L)
• Replacing t with x/L gives: f(x) ≤ (x/L)·f(L)

The line y = (f(L)/L)x is the chord connecting (0,0) and (L, f(L)), which forms the hypotenuse of our right triangle. Hence, the entire curve lies at or below this hypotenuse.

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3. Incorporating Discrete Slope Data

Suppose you know the slope (in degrees) of the curve at every 2‑foot increment along the horizontal axis. Denote these measurements by:

θ₀, θ₁, θ₂, ..., θₙ

where:

• θ₀ = 0°,
• xᵢ = 2i feet (for i = 0, 1, ..., n),
• L = 2n feet.

For a short horizontal segment of 2 feet, the vertical change is approximately:

Δyᵢ ≈ 2 tan(θᵢ)

Thus, the cumulative vertical rise of the curve can be approximated by summing over these segments:

f(L) ≈ Σᵢ₌₀ⁿ⁻¹ 2 tan(θᵢ)

In the limit of infinitely small increments, this sum converges to the exact value:

f(L) = ∫₀ᴸ f'(x) dx

Once you have the measurements:

• L = 2n feet (horizontal distance) and
• f(L) (total vertical rise),

you can calculate:

• the hypotenuse h = √(L² + f(L)²),
• and the angle α = arctan(f(L)/L).

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4. Summary of the Proof

1. Curve Definition and Convexity:
   The curve y=f(x) starts at (0,0) with a 0° slope and has a continuously increasing slope up to an angle θ_L ≤ 90°. This implies that f'(x) is non‑decreasing, rendering the function convex.

2. Bounding the Curve:
   Convexity ensures that for all x in [0,L], f(x) ≤ (x/L)·f(L), meaning the curve is entirely contained within the straight line (or chord) joining (0,0) and (L, f(L)).

3. Constructing the Encompassing Triangle:
   The chord serves as the hypotenuse of a right triangle with horizontal leg L and vertical leg f(L). Consequently:

   h = √(L² + f(L)²) and α = arctan(f(L)/L)
   

4. Utilizing Discrete Slope Measurements:
   Knowing the slope at every 2‑foot increment lets us approximate the total vertical rise f(L) via:

   f(L) ≈ Σᵢ₌₀ⁿ⁻¹ 2 tan(θᵢ)
   

   which in turn allows us to compute the exact dimensions of the right triangle.

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5. Conclusion

We have proven that any curve with a starting 0° slope that smoothly and convexly increases to at most 90° can be encompassed within a right triangle. By harnessing discrete slope measurements at regular intervals, one can approximate the vertical rise of the curve and, ultimately, calculate both the length and the angle of the hypotenuse of the corresponding right triangle:

h = √(L² + f(L)²) and α = arctan(f(L)/L)

This elegant method bridges calculus and geometry, providing a practical way to understand and enclose curved shapes using simple geometric constructs.