LAGRANGE MULTIPLIERS AND THE CALCULUS OF VARIATION IN GAME DESIGN PAUL BOUTHELLIER DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE UNIVERSITY OF PITTSBURGH-TITUSVILLE TITUSVILLE, PA 16354 [email protected] FINDING THE QUICKEST PATH INITIAL PROBLEM • Can travel at a speed of si on level li • Find the path C which minimizies the time it takes to get from (0, 0) to (1000, 1000) • Consider general speeds • Consider falling under gravity alone • Falling under gravity with an atmosphere • Dealing with obstacles • R2, R3, R4 TECHNIQUES • Lagrange Multipliers • Euler-Lagrange Equation • Beltrami Equation • Numerical Analysis THE LAGRANGE MULTIPLIER METHOD The Lagrange Multiplier Method + 1, 2, … , = 2 2 � =1 THE LAGRANGE MULTIPLIER METHOD The Lagrange Multiplier Method + 1, 2, … , = 2 2 � =1 + 1, 2, … , = 2 2 � =1 Lagrange Multipliers: Solving for the di d 1 is 9.717592076941 d 2 is 19.716469280426 d 3 is 30.320551487255 d 4 is 41.955199508211 d 5 is 55.250555011642 d 6 is 71.258726625095 d 7 is 91.996416566653 d 8 is 122.145576401652 d 9 is 176.855999289317 d 10 is 380.782915477145 Total time is 35.777931135649 Total distance is 1542.980226242951 EXAMPLE II d 1 is 19.008231985751 d 2 is 40.261059108635 d 3 is 67.630647087404 d 4 is 112.345176987363 d 5 is 260.754884222299 d 6 is 260.754884222299 d 7 is 112.345176987363 d 8 is 67.630647087404 d 9 is 40.261059108635 d 10 is 19.008231985751 Total time is 57.877432196461 Total distance is 1519.980402465238 FIRST COUSIN: THE BRACHISTOCHRONE PROBLEM Bernoulli (1696): Given two points A and B, find the path along which an object would slide (without friction) in the shortest time from A to B, if it starts at A in rest and is only accelerated by gravity. Solving The Brachistochrone Problem Where y(x) is the solution which minimizes the time T where To make T stationary use the Euler-Lagrange equation However, as F does not contain x can use the special case-The Beltrami Identity = − ′ ′ Solving Beltrami's Identity Using separable equations, substitutions, and some hand-waving we get x(t)=a(t-sin(t)) y(t)=a(1-cos(t)) Where the range t=0..tE comes from: Time along curve Example Letting xE=yE=1000 Brachistochrone: Lagrange Multiplier Method (n=1000 partitions) x(t)=572.917(t-sin(t)) d1=20.24 y(t)=572.917(1-cos(t)) d2=58.93 d3=111.74 t=0..2.412 d4=178.08 d5=258.56 d6=354.75 d7=469.44 d8=607.44 d9=777.67 d10=1000 Atmospheric Model of a Skydiver m=mass of the skydiver g=gravity d=density of the air A=cross-sectional area CD=coefficient of drag v=skydiver's velocity = tanh(t ) 2 2 Graph of v(t) Speed in mph Approximate by: Example-Dealing with Terminal Velocity Break the Problem Into Two Pieces Minimum travel time is: 31.889174788418.