Helge Rhodin CPSC 427 Video Game Programming Transformations for Skeleton Kinematics 1 © Alla Sheffer, Helge Rhodin Recap: Fun to play? https://www.androidauthority.com/level-design-mobile-games-developers- make-games-fun-661877/ 2 © Alla Sheffer, Helge Rhodin Recap: Indirect relationships Value of a piece • It is not possible to get a knight for 3 pawns • But one can ‘trade’ pieces • A currency How to determine the value? • Ask the users • Auction house 3 © Alla Sheffer, Helge Rhodin Recap: Relationships • Linear relations • Exponential relations • Triangular relationship • 1, 3, 6, 10, 15, 21, 28, … • The difference increases linearly • The function has quadratic complexity • Periodic relations 4 © Alla Sheffer, Helge Rhodin Asymptotic analysis? • Linear * linear? • Linear + linear? • Linear + exponential? • Linear * exponential? 5 © Alla Sheffer, Helge Rhodin Numerical Methods - Optimization • Iterative optimizers • Single variable? Exponential Target function • Multiple variables? Linear approx. • Gradient descent? Constant approx. Lecture 14: https://youtu.be/ZNsNZOnrM50 • Balancing demo starts at 1h20 • Optimizer used at ~ 1h30 6 © Alla Sheffer, Helge Rhodin Difficulties: • Placement of towers changes the time damage is dealt • Path of enemies can be hindered to increase time ➢ Measure during playtest ➢ cross-play • Some enemies are resistant to fire/magic/…? • kind of a periodic feature 7 © Alla Sheffer, Helge Rhodin Counter Measures • Transitive Mechanics • Repair costs • Consumables (food, potions, …) • Tax 8 © Alla Sheffer, Helge Rhodin A/B Testing Testing two variants of your game (with and w/o a feature) • randomized participants (same pool) • with respect to a measurable objective (e.g., clicks on website) Related to • two-sample hypothesis testing • Clinical tests, e.g., testing of a COVID-19 vaccine • Placebo effect 9 https://www.optimizely.com/optimization-glossary/ab-testing/ © Alla Sheffer, Helge Rhodin Logistics 10 © Alla Sheffer, Helge Rhodin M4 updated requirement User study Carefully balance one aspect of your game (e.g., movement-speed, health points, strength, bonus,…). • Report on the theoretical analysis • Change log with testing 11 © Alla Sheffer, Helge Rhodin M4 Advanced shaders? • Use ShaderToy.com • Demo in Lecture 4 https://youtu.be/S97-fYMv4Xk 45 minute mark 12 © Alla Sheffer, Helge Rhodin Upcoming lectures: Tuesday: • Working in teams • Supported by TA Thursday: • Last formal lecture: • Skeleton Animation continued • Summary and Outlook Tuesday (April 13.): • M4 grading with TAs Monday (April 19.): • Final cross play, including industry jury and awards 13 © Alla Sheffer, Helge Rhodin Final cross play Room Purpose Timeline: 1 Team 1 – open cross-play 7 – 8 pm: scheduled (10 minutes) cross play 2 Team 2 – open cross-play 7 – 8:30 pm: open cross play (those not with jury) … … 8 – 8:30 pm: jury votes, student votes are counted 8:30 – 9 pm: awards! 12 Team 12 – open cross-play 13 Jury 1 (Skybox) – scheduled 10 min slots Communication: 14 Jury 2 (EA) – scheduled 10 min slots On Slack to reach beyond zoom rooms … … • OK to include jury? 20 Overflow – breakout for open cross-play 21 Overflow – breakout for open cross-play Presence: … … • Mandatory! 14 © Alla Sheffer, Helge Rhodin Today Becoming an expert on transformation • Mental picture • Math • Practical examples Composite transformations • Articulated skeleton motion • Skeleton animation 15 © Alla Sheffer, Helge Rhodin Recap: Transformations Lecture 3, 20 minute mark • https://youtu.be/l9xQUxJT7fg 16 © Alla Sheffer, Helge Rhodin From local object to camera coordinates Camera Camera World World coordinates coordinates coordinates coordinates y y x x x x y y object -> world world -> camera object -> camera transform projection projection * transform 17 © Alla Sheffer, Helge Rhodin Recap: GLSL Vertex shader The OpenGL Shading Language (GLSL) • Syntax similar to the C programming language • Build-in vector operations x and y coordinates • functionality as the GLM library our assignment template uses of a vec2, vec3 or vec4 world object void main() -> camera -> world { // Transforming The Vertex vec3 out_pos = projection * transform * vec3(in_pos.xy, 1.0); gl_Position = vec4(out_pos.xy, in_pos.z, 1.0); } float vector of 3 (vec3) and 4 (vec4) floats (32 bit) 18 © Alla Sheffer, Helge Rhodin Affine transformations • Linear transformations + translations • Can be expressed as 2x2 matrix + 2 vector • E.g. scale+ translation: x' x'2 02 00xx Ttxx = + y'=y'0 20 02yy +Tty z' 0 0 1 z 0 © Alla Sheffer, Helge Rhodin Modeling Transformation Adding a third coordinate x' x'2 02 00xx Ttxx = + y'=y'0 20 02yy +Tty z' 0 0 1 z 0 Affine transformations are now linear • one 3x3 matrix can express: 2D rotation, scale, shear, and translation © Alla Sheffer, Helge Rhodin Forward transformations • Given position, scale, angle • Compute transformation matrix • Transform the object • Examples? • Later: Inverse transformations 21 © Alla Sheffer, Helge Rhodin Parallax scrolling background Further objects • are smaller • move slower Multiple layers! Weak perspective projection • Approximates perspective projection 22 © Alla Sheffer, Helge Rhodin Parallax scrolling background Ingredients: • Multiple layers • One sprite per layer x' 2 0 0 x t • One transformation per layer x • Scale y' down= 0 with2 distance0 y + ty • Move z' inversely 0 0 proportional1 z to0 the distance 23 © Alla Sheffer, Helge Rhodin 24 © Alla Sheffer, Helge Rhodin TRANSFORMATION COMPOSITION p x • What operation rotates XY by around P = ? py P P © Alla Sheffer, Helge Rhodin TRANSFORMATION COMPOSITION p x • What operation rotates XY by around P = ? py • Answer: • Translate P to origin P P © Alla Sheffer, Helge Rhodin TRANSFORMATION COMPOSITION p x • What operation rotates XY by 휃 < 0 around P = ? py • Answer: • Translate P to origin • Rotate around origin by • Translate back P P © Alla Sheffer, Helge Rhodin TRANSFORMATION COMPOSITION ( p , p ) (− px ,− p y ) T x y RT (V ) 1 0 p cos − sin 01 0 − p v x x x = 0 1 p sin cos 00 1 − p v y y y 0 0 1 0 0 10 0 1 1 © Alla Sheffer, Helge Rhodin two interpretations cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATES cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATES cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATES cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME Columns are new basis vectors (and new origin)! cos 휃 − sin 휃 푝푥 ⋅ (1−cos 휃) + 푝푦 ⋅ sin휃 푣푥 sin 휃 cos 휃 푝푦 ⋅ (1−cos 휃) + 푝푥 ⋅ sin휃 푣푦 0 0 1 1 P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME 풗풙 (풑 ,풑 ) 휽 (−풑 ,−풑 ) 푻 풙 풚 푹 푻 풙 풚 풗풚 ퟏ P P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME 푣푥 (풑 ,풑 ) 휃 (−푝 ,−푝 ) 푻 풙 풚 푅 푇 푥 푦 푣푦 1 P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME 푣푥 (풑 ,풑 ) 휃 (−푝 ,−푝 ) 푻 풙 풚 푅 푇 푥 푦 푣푦 1 P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME 푣푥 (푝 ,푝 ) 휽 (−푝 ,−푝 ) 푇 푥 푦 푹 푇 푥 푦 푣푦 1 P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME 푣푥 (푝 ,푝 ) 휃 (−풑 ,−풑 ) 푇 푥 푦 푅 푻 풙 풚 푣푦 1 P © Alla Sheffer, Helge Rhodin TRANSFORMING COORDINATE FRAME World Coordinate Object Coordinate Frame Frame 푣′푥 푣푥 (푝푥,푝푦) 휃 (−푝푥,−푝푦) 푣′푦 = 푇 푅 푇 푣푦 1 1 P P © Alla Sheffer, Helge Rhodin TWO INTERPRETATIONS OF COMPOSITE World Coordinate Object Coordinate Frame Frame 푣′푥 푣푥 (푝푥,푝푦) 휃 (−푝푥,−푝푦) 푣′푦 = 푇 푅 푇 푣푦 1 1 1) read from inside-out as transformation of object 2) read from outside-in as transformation of the coordinate frame © Alla Sheffer, Helge Rhodin Transformation Hierarchies © Alla Sheffer, Helge Rhodin Transformation Hierarchies Scenes have multiple coordinate systems • Often strongly related • Parts of the body • Object on top of each other • Next to each other… Independent definition is bug prone Solution: Transformation Hierarchies © Alla Sheffer, Helge Rhodin Transformation Hierarchy Examples 2 4 5 3 1 y x © Alla Sheffer, Helge Rhodin Transformation Hierarchy Examples 푀1 = 푇푟(푥,푦) ⋅ 푅표푡휃1 2 푀2 = 푀1 ⋅ 푇푟(2.5,5.5) ⋅ 푅표푡 휃2 4 푀3 = 푀2 ⋅ 푇푟(0,−3.5) ⋅ 푅표푡 휃3 5 3 1 y x © Alla Sheffer, Helge Rhodin Transformation Hierarchies world torso LUleg RUleg LUarm RUarm head LLleg RLleg LLarm RLarm Lfoot Rfoot Lhand Rhand rot(z, ) trans(0.30,0,0) © Alla Sheffer, Helge Rhodin Transformation Hierarchy Quiz M.setIdentity(); M = M*Translation(4,1,0); M = M*Rotation(pi/4,0,0,1); House.matrix = M; Which color house will we draw? A. Red B. Blue C. Green FW D. Orange E. Purple © Alla Sheffer, Helge Rhodin Hierarchical Modeling Advantages • Define object once, instantiate multiple copies • Transformation parameters often good control knobs • Maintain structural constraints if well-designed Limitations • Expressivity: not always the best controls • Can’t do closed kinematic chains • E.g., how to keep a hand on the hip? © Alla Sheffer, Helge Rhodin Forward vs.