#363 - Bézier Curves
A cubic Bézier curve is defined by four points: \(P_0, P_1, P_2,\) and \(P_3\).
The curve is constructed as follows:
On the segments \(P_0 P_1\), \(P_1 P_2\), and \(P_2 P_3\) the points \(Q_0, Q_1,\) and \(Q_2\) are drawn such that \(\dfrac{P_0 Q_0}{P_0 P_1} = \dfrac{P_1 Q_1}{P_1 P_2} = \dfrac{P_2 Q_2}{P_2 P_3} = t\), with \(t\) in \([0, 1]\).
On the segments \(Q_0 Q_1\) and \(Q_1 Q_2\) the points \(R_0\) and \(R_1\) are drawn such that
\(\dfrac{Q_0 R_0}{Q_0 Q_1} = \dfrac{Q_1 R_1}{Q_1 Q_2} = t\) for the same value of \(t\).
On the segment \(R_0 R_1\) the point \(B\) is drawn such that \(\dfrac{R_0 B}{R_0 R_1} = t\) for the same value of \(t\).
The Bézier curve defined by the points \(P_0, P_1, P_2, P_3\) is the locus of \(B\) as \(Q_0\) takes all possible positions on the segment \(P_0 P_1\).
(Please note that for all points the value of \(t\) is the same.)
From the construction it is clear that the Bézier curve will be tangent to the segments \(P_0 P_1\) in \(P_0\) and \(P_2 P_3\) in \(P_3\).
A cubic Bézier curve with \(P_0 = (1, 0), P_1 = (1, v), P_2 = (v, 1),\) and \(P_3 = (0, 1)\) is used to approximate a quarter circle.
The value \(v \gt 0\) is chosen such that the area enclosed by the lines \(O P_0, OP_3\) and the curve is equal to \(\dfrac{\pi}{4}\) (the area of the quarter circle).
By how many percent does the length of the curve differ from the length of the quarter circle?
That is, if \(L\) is the length of the curve, calculate \(100 \times \dfrac{L - \frac{\pi}{2}}{\frac{\pi}{2}}\)
Give your answer rounded to 10 digits behind the decimal point.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=363. Published Sunday, 18th December 2011, 10:00 am. Solved by 1,336 members at time of mirroring.
Why this is useful
Numerical Computing. Precision, conditioning, and numerical iteration transfer directly to pricing engines and model calibration (Phases 5, 13, 15).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.1 Sets, Functions, and Relations · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 2.1 Functions, Limits, and Continuity · 2.4 Taylor Series and Local Approximation · 3.1 Vectors, Multivariable Functions, and Level Sets · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Recommended stepping-stone problems: #587 · #504 · #577
Concepts: algebra computational-geometry geometry numerical-methods
Likely techniques: backtracking memoization
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Understand the problem
- What exactly is the input to problem 363? Is it a bound (100), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a real number quoted to a stated precision, so the whole computation must control rounding error. Required format: Give your answer rounded to 10 digits behind the decimal point.
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What constraint does the bound 100 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 100?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 100 and the cost of testing one.
- Which geometry fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says geometry / backtracking - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 100, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it - then run it. A surprise here is worth more than an hour of debugging later.
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Optimization
You have a correct answer. That is the start of the learning, not the end.
- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
- Reduce memory. Can you stream, or keep only the last k states?
- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
- Prove the optimized version computes the same thing.
- Compare two implementations and time them.
Explain it
Which step of your solution were you least confident about, and what evidence would settle it?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the geometry structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the backtracking idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound 100 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'backtracking' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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You reach Mastered only when you have solved it, rated yourself at least Solid across the dimensions, and written a real explanation.
Confidence
Low confidence schedules this problem for spaced review, even if you solved it.
Mastery check
- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #587 · #504 · #577
- Transfer: where else does this technique appear? Name a lesson and a real computational setting.
- Spaced re-attempt: come back after the review interval and re-solve it with no hints.