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CSCE 552 Fall 2012. Math, Physics and Collision Detection. By Jijun Tang. Homework #2. Major use cases of your system Due on Wednesday Oct 17 th , before class. Use Cases. Use Case Name: Place Order Actors:
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CSCE 552 Fall 2012 Math, Physics and Collision Detection By Jijun Tang
Homework #2 • Major use cases of your system • Due on Wednesday Oct 17th, before class.
Use Cases • Use Case Name: Place Order • Actors: • Registered Shopper (Has an existing account, possibly with billing and shipping information) • Fulfillment System (processes orders for delivery to customers) • Billing System (bills customers for orders that have been placed) • Triggers: • The user indicates that she wants to purchase items that she has selected. • Preconditions: • User has selected the items to be purchased. • Post-conditions: • The order will be placed in the system. • The user will have a tracking ID for the order. • The user will know the estimated delivery date for the order. • Flow: • The user will indicate that she wants to order the items that have already been selected. • The system will present the billing and shipping information that the user previously stored. • The user will confirm that the existing billing and shipping information should be used for this order. • The system will present the amount that the order will cost, including applicable taxes and shipping charges. • The user will confirm that the order information is accurate. • The system will provide the user with a tracking ID for the order. • The system will submit the order to the fulfillment system for evaluation. • The fulfillment system will provide the system with an estimated delivery date. • The system will present the estimated delivery date to the user. • The user will indicate that the order should be placed. • The system will request that the billing system should charge the user for the order. • The billing system will confirm that the charge has been placed for the order. • The system will submit the order to the fulfillment system for processing. • The fulfillment system will confirm that the order is being processed. • The system will indicate to the user that the user has been charged for the order. • The system will indicate to the user that the order has been placed. • The user will exit the system.
Real-time Physics in Game at Runtime: • Enables the emergent behavior that provides player a richer game experience • Potential to provide full cost savings to developer/publisher • Difficult • May require significant upgrade of game engine • May require significant update of asset creation pipelines • May require special training for modelers, animators, and level designers • Licensing an existing engine may significantly increase third party middleware costs
Particle Position • Location of Particle in World Space • SI Units: meters (m) • Changes over time when object moves
Particle Velocity and Acceleration • Velocity (SI units: m/s) • First time derivative of position: • Acceleration (SI units: m/s2) • First time derivative of velocity • Second time derivative of position
Newton’s 2nd Law of Motion • Paraphrased –“An object’s change in velocity is proportional to an applied force” • The Classic Equation: • m = mass (SI units: kilograms, kg) • F(t) = force (SI units: Newtons)
Concrete Example: Target Practice Projectile Launch Position, pinit Target
Finite Difference Methods-I • The Explicit Euler Integrator: • Properties of object are stored in a state vector, S • Use the above integrator equation to incrementally update S over time as game progresses • Must keep track of prior value of S in order to compute the new • For Explicit Euler, one choice of state and state derivative for particle:
Finite Difference Methods-II • The Verlet Integrator: • Must store state at two prior time steps, S(t) and S(t-Dt) • Uses second derivative of state instead of the first • Valid for constant time step only (as shown above) • For Verlet, choice of state and state derivative for a particle:
Errors Exact Euler
Aerodynamic Drag S: projected front area CD: drag coefficient
What is Collision Detection • A fundamental problem in computer games, computer animation, physically-based modeling, geometric modeling, and robotics. • Including algorithms: • To check for collision, i.e. intersection, of two given objects • To calculate trajectories, impact times and impact points in a physical simulation.
Collision Detection • Complicated for two reasons • Geometry is typically very complex, potentially requiring expensive testing • Naïve solution is O(n2) time complexity, since every object can potentially collide with every other object • Two basic techniques • Overlap testing: Detects whether a collision has already occurred • Intersection testing: Predicts whether a collision will occur in the future
Overlap Testing (a posteriori) • Overlap testing: Detects whether a collision has already occurred, sometime is referred as a posteriori • Facts • Most common technique used in games • Exhibits more error than intersection testing • Concept • For every (small) simulation step, test every pair of objects to see if they overlap • Easy for simple volumes like spheres, harder for polygonal models
Overlap Testing Results • Useful results of detected collision • Pairs of objects will have collision • Time of collision to take place • Collision normal vector • Collision time calculated by moving object back in time • until right before collision • Bisection is an effective technique
Bisect Testing: Iteration V Time right before the collision
Overlap Testing: Limitations Fails with objects that move too fast • Thin glass vs. bulltes • Unlikely to catch time slice during overlap
Solution for This Limitation • Speed of the fastest object multiplies the time step should be smaller than the smallest objects in the scene • Possible solutions • Design constraint on speed of objects: hard to apply without affecting the play • Reduce simulation step size: too expensive
Intersection Testing (a priori) • Predict future collisions • When predicted: • Move simulation to time of collision • Resolve collision • Simulate remaining time step
Intersection Testing:Swept Geometry • Extrude geometry in direction of movement • Swept sphere turns into a “capsule” shape
Special Cases • No collision: • B2 = 0: both objects are stationary, or they are traveling at parallel • When will collision occur?
Intersection Testing:When to Collide • Smallest distance ever separating two spheres: • If there is a collision
Intersection Testing:Limitations • Issue with networked games • Future predictions rely on exact state of world at present time • Due to packet latency, current state not always coherent • Assumes constant velocity and zero acceleration over simulation step • Has implications for physics model and choice of integrator
Dealing with Complexity Two issues 1. Complex geometry must be simplified 2. Reduce number of object pair tests
Simplified Geometry Approximate complex objects with simpler geometry, like this ellipsoid or bounding boxes
Minkowski Sum By taking the Minkowski Sum of two complex volumes and creating a new volume, overlap can be found by testing if a single point is within the new volume
Bounding Volumes • Bounding volume is a simple geometric shape • Completely encapsulates object • If no collision with bounding volume, no more testing is required • Common bounding volumes • Sphere • Box
Using Bounding Box in Game • Complex objects can have multiple bounding boxes • Human object can have one big bounding box for the whole body • Human object can have one bounding box per limb, head, etc • Bounding box can be hierarchical: • Test the big first • if possible collision, test the smaller ones
Reduce Number of Detections O(n) Time Complexity can be achieved. One solution is to partition space
Achieving O(n) Time Complexity Another solution is the plane sweep algorithm Requires (re-)sorting in x (y) coordinate
