arc length - 2024 PLAYTIME

Question

Answer the following about the Frenet–Serret (“freh-NAY sir-AY”) formulas, which describe the dynamics of the TNB frame along a curve. For 10 points each:

[10m] The Frenet–Serret formulas give expressions for the derivatives of the T, N, and B vectors with respect to this quantity, denoted s. For a function f, the [read slowly] “integral of the square root of 1 plus f-prime squared” gives this quantity, which is finite for rectifiable curves.

ANSWER: arc length [prompt on length]

[10h] The Frenet–Serret formulas involve two constants: the curvature, denoted kappa, and this other quantity, computed as the negative dot product of N and B prime. This quantity measures how quickly a curve is escaping its osculating plane.

ANSWER: torsion [prompt on tau]

[10e] The first and third formulas produce multiples of the normal vector N, which is named for having this property with respect to the tangent vector. Two vectors have this property if they meet at an angle of 90 degrees.

ANSWER: orthogonal [or perpendicular; prompt on normal]

<JC, Other Science - Math>

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Summary


2025 PLAYTIME @ CWRU	12/10/2024	Y	5	16.00	100%	60%	0%
2025 PLAYTIME @ Claremont	12/10/2024	Y	5	16.00	100%	40%	20%
2025 PLAYTIME @ Florida	12/10/2024	Y	2	10.00	100%	0%	0%
2025 PLAYTIME @ Florida	12/10/2024	Y	1	30.00	100%	100%	100%
2024 PLAYTIME @ Harvard	04/26/2025	Y	4	22.50	100%	100%	25%
2024 PLAYTIME @ JHU	10/26/2024	Y	8	15.00	100%	50%	0%
2025 PLAYTIME @ Northwest	12/10/2024	Y	4	15.00	75%	50%	25%
2025 PLAYTIME Online	04/26/2025	Y	8	15.00	100%	38%	13%
2025 PLAYTIME @ Sheffield	12/10/2024	Y	6	16.67	100%	50%	17%
2025 PLAYTIME @ Sheffield	12/10/2024	Y	1	10.00	100%	0%	0%
2024 PLAYTIME @ UBC	10/05/2024	Y	8	17.50	88%	63%	25%
2024 PLAYTIME @ Virginia Tech	07/11/2025	Y	3	13.33	100%	33%	0%
2024 PLAYTIME @ WashU	10/12/2024	Y	6	20.00	100%	67%	33%

Data


Michigan A	Pittsburgh	10	0	10	20
Michigan C	Ohio State A	0	0	10	10
UCLA B	Caltech	0	0	10	10
UCLA A	Claremont A	10	10	10	30
UCSD	Claremont B	10	0	10	20
Art of Premature Signaling	P.C.‚ Daniel‚ Michel‚ and Jason B	0	0	10	10
girl 15 whale i mean they	UCSB	0	0	10	10
Billy's Bastion	Project2025BC: Make Assyria Great Again	0	0	10	10
Edwin Laundrorat's Long-Lost Man Unclothès Rat Degrosses	Keeping Up With the Cardassians	0	0	10	10
Too much from you lately	Qin Shi Huang's Immortality Poison	10	10	10	30
Lexington	Belmont	10	0	10	20
Brown	RIT	10	10	10	30
Harvard	BU	10	0	10	20
MIT	Brandeis	10	0	10	20
Michigan B	Ohio State B	0	0	10	10
George Mason A	CWRU	0	0	10	10
Georgetown	George Mason B	10	0	10	20
NYU C	George Mason C	10	0	10	20
Maryland A	George Washington	10	0	10	20
Haverford A	Johns Hopkins A	0	0	10	10
Haverford B	NYU B	0	0	10	10
Johns Hopkins B	Catholic University of America	10	0	10	20
Maryland B	NYU A	0	0	10	10
Michigan D	Ohio State C	10	0	10	20
Arizona State University A	Carleton	10	10	10	30
Arizona State University B	Alberta	10	0	10	20
UW B	UW A	0	0	10	10
Washington Connections	South Dakota School of Mines	0	0	0	0
Ohio State D	Oberlin	10	0	10	20
Four Generalists in Search of a Power	Auburn 2026	0	0	10	10
Quatumvirate	Goober Gang	10	0	10	20
Gruh	Mahjonng Masters	0	0	10	10
Monsieur Hulot's Academic Team	Mike Dukakis Tank Battalion	10	10	10	30
Trip-Hop Quizzer Tricky	Free Agents 1	10	0	10	20
Egypt Airlines Haters	evil women enjoyers	0	0	10	10
mongeese	MSJ	0	0	10	10
we need new names	Illinois Orange	0	0	10	10
Imperial A.	Imperial B	0	0	10	10
Birmingham	Manchester	0	0	10	10
Sheffield A	Edinburgh	10	10	10	30
Cambridge	Sheffield B	10	0	10	20
Warwick A	LSE	10	0	10	20
Warwick B	Lancaster	0	0	10	10
Warwick C	Sheffield C	0	0	10	10
Waterloo C	Guelph	10	10	0	20
UBC	Ottawa A	0	0	10	10
Waterloo B	Toronto A	10	0	10	20
Toronto Metropolitan	Toronto B	10	0	10	20
Toronto C	Simon Fraser	0	0	10	10
Toronto D	Toronto E	10	0	10	20
Waterloo A	McGill A	10	0	10	20
Mac A	McGill B	0	10	10	20
Even in Blacksburg There I Am	Somehow Paula Deen Returned	0	0	10	10
4 quizbowl pros go Virginia on North Carolina	Adrianne Lenker 4 Nobel Lit Prize 2K65	10	0	10	20
Master and Margaritaville	Ashurbanipal’s Drag Race	0	0	10	10
Chicago B	WashU Squirtle	10	10	10	30
Illinois A	Chicago A	10	10	10	30
Illinois B	Illinois C	0	0	10	10
Iowa State	Truman State	10	0	10	20
Missouri	WashU Bulbasaur	10	0	10	20
WashU Charmander	SIUE	0	0	10	10