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Computational Geometry & Collision detection. Vectors, Dot Product, Cross Product, Basic Collision Detection. George Georgiev. Telerik Corporation. www.telerik.com. Table of Contents. Vectors Extended revision The vector dot product The vector cross product Collision detection
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Computational Geometry & Collision detection Vectors, Dot Product, Cross Product, Basic Collision Detection George Georgiev Telerik Corporation www.telerik.com
Table of Contents • Vectors • Extended revision • The vector dot product • The vector cross product • Collision detection • In Game programming • Sphere collision • Bounding volumes • AABBs
Vectors Revision, Normals, Projections
Vectors – revision • Ordered sequences of numbers • OA (6, 10, 18) – 3-dimensional • OA (6, 10) – 2-dimensional • OA (6, 10, 18, -5) – 4-dimensional • Have magnitude and direction A
Vectors – revision • No location • Wherever you need them • Can represent points in space • Points are vectors with a beginning at the coordinate system center • Example: • Point A(5, 10) describes the location (5, 10) • Vector U(5, 10), beginning at (0, 0), describes ‘the path’ to the location (5, 10)
Vectors – revision • All vectors on the same line are called collinear • Can be derived by scaling any vector on the line • E.g.: A(2, 1), B(3, 1.5), C(-1, -0.5) are collinear • Two vectors, which are not collinear, lie on a plane and are called coplanar • => Two non-collinear vectors define a plane • Three vectors, which are not coplanar, define a space
Vectors – revision • Collinear vectors: • Coplanar vectors:
Vectors – revision • Vectors defining a 3D vector space
Vectors – revision • Perpendicular vectors • Constitute a right angle • Deriving a vector, perpendicular to a given one: • Swap two of the coordinates of the given vector (one of the swapped coordinates can’t be zero) • Multiply ONE of the swapped coordinates by -1 • Example: • A (5, 10) given => A’(-10, 5) is perpendicular to A • V (3, 4, -1) given => V’(3, 1, 4) is perpendicular to V
Vectors – revision • Normal vectors to a surface • Constitute a right angle with flat surfaces • Perpendicular to at least two non-collinear vectors on the plane • Constitute a right angle with the tangent to curved surfaces
Vectors – revision • Projection of a vector on another vector
Vector Dot Product Definition, Application, Importance
Vector dot product • Dot Product (a.k.a. scalar product) • Take two equal-length sequences • e.g. sequence A (5, 6) and sequence B (-3, 2) • Multiply each element of A with each element of B • A [i] * B [i] • Add the products • Dot Product(A, B) = A[0] * B[0] + A[1] * B[1] + … + A[i] * B[i] + … + A[n-1] * B[n-1]
Vector dot product • Dot Product (2) • Example: • A (5, 6) B (-3, 2) = 5 * (-3) + 6 * 2 = -15 + 12 = -3 • Result • A scalar number
Vector dot product • Dot product of coordinate vectors • Take two vectors of equal dimensions • Apply the dot product to their coordinates • 2D Example: • A(1, 2) . B(-1, 1) = 1*(-1) + 2*1 = 1 • 3D Example: • A(1, 2, -1) . B(-1, 1, 5) = 1*(-1) + 2*1 + (-1) * 5 = -4 • Simple as that
Vector dot product • Meaning in Euclidean geometry • If A(x1, y1, …), B(x2, y2, …) are vectors • theta is the angle, in radians, between A and B • Dot Product (A, B) = A . B = = |A|*|B|*cos(theta) • Applies to all dimensions (1D, 2D, 3D, 4D, … nD)
Vector dot product • Meaning in Euclidean geometry (2) • If U and V are unit vectors, then U . V = • cosine of the angle between U and V • the oriented length of the projection of U on V • If U and V are non-unit vectors • ( U . V ) divided by |U|*|V| = cosine of the angle between U and V • ( U . V ) divided by |V| = the oriented length of the projection of U on V
Vector dot product • Consequences • If A . B > 0, A and B are in the same half-space • If A . B = 0, A and B are perpendicular • If A . B < 0, A and B are in different half-spaces • Applications • Calculating angles • Calculating projections • Calculating lights • Etc…
Dot Product Computation Live Demo
Vector Cross Product Definition, Features, Application
Vector cross product • Cross product • Operates on vectors with up to 3 dimensions • Forms a determinant of a matrix of the vectors • Result – depends on the dimension • In 2D – a scalar number (1D) • In 3D – a vector (3D) • Not defined for 1D and dimensions higher than 3
2D Vector cross product • 2D Cross product • Take the vectors U(x1, y1) and V(x2, y2) • Multiply their coordinates across and subtract: • U(x1, y1) x V(x2, y2)= (x1 * y2) – (x2 * y1) • Result • A scalar number
2D Vector cross product • Scalar meaning in Euclidean geometry • If U(x1, y1) and V(x2, y2) are 2D vectors • theta is the angle between U and V • Cross Product (U, V) = U x V = = |U| * |V| * sin(theta) • |U| and |V| denote the length of U and V • Applies to 2D and 3D
2D Vector cross product • Scalar meaning in Euclidean geometry (2) • For every two 2D vectors U and V • U x V = the oriented face of the parallelogram, defined by U and V • For every three 2D points A, B and C • If U x V = 0, then A, B and C are collinear • If U x V > 0, then A, B and C constitute a ‘left turn’ • If U x V < 0, then A, B and C constitute a ‘right turn’
2D Vector cross product • Applications • Graham scan (2D convex hull) • Easy polygon area computation • Cross product divided by two equals oriented (signed) triangle area • 2D orientation • ‘left’ and ‘right’ turns
2D Cross Product Computation Live Demo
3D Vector cross product • 3D Cross product • Take two 3D vectors U(x1, y1, z1) and V(x2, y2, z2) • Calculate the following 3 coordinates • x3 = y1*z2 – y2*z1 • y3 = z1*x2 – z2*x1 • z3 = x1*y2 – x2*y1 • Result • A 3D vector with coordinates (x3, y3, z3)
3D Vector cross product • Meaning in Euclidean geometry • The magnitude • Always positive (length of the vector) • Has the unsigned properties of the 2D dot product • The vector • Perpendicular to the initial vectors U and V • Normal to the plane defined by U and V • Direction determined by the right-hand rule
3D Vector cross product • The right-hand rule • Index finger points in direction of first vector (a) • Middle finger points in direction of second vector (b) • Thumb points up in direction of the result of a x b
3D Vector cross product • Unpredictable results occur with • Cross product of two collinear vectors • Cross product with a zero-vector • Applications • Calculating normals to surfaces • Calculating torque (physics)
3D Cross Product Computation Live Demo
Computational geometry ? ? Questions? ? ? ? ? ? ? ? ? ?
Collision detection Basics, Methods, Problems, Optimization
Collision detection • Collisions in Game programming • Any intersection of two objects’ geometry • Raise events in some form • Usually the main part in games • Collision response – deals with collision events
Collision detection • Collision objects • Can raise collision events • Types • Spheres • Cylinders • Boxes • Cones • Height fields • Triangle meshes
Collision detection • Sphere-sphere collision • Easiest to detect • Used in • particle systems • low-accuracy collision detection • Collision occurrence: • Center-center distance less than sum of radiuses • Optimization • Avoid computation of square root
Sphere-sphere collision detection Live Demo
Collision detection • Triangle meshes collision • Very accurate • Programmatically heavy • Computation heavy (n2) • Rarely needed
Collision detection • Collision detection in Game programming • Combines several collision models • Uses bounding volumes • Uses optimizations • Axis-sweep • Lower accuracy in favor of speed
Collision detection • Bounding volumes • Easy to check for collisions • Spheres • Boxes • Cylinders, etc. • Contain high-triangle-count meshes • Tested for collision before the contained objects • If the bounding volume doesn’t collide, then the mesh doesn’t collide
Collision detection • Bounding sphere • Orientation-independent • Center – mesh’s center • Radius • distance from mesh center to farthest vertex • Effective for • convex, oval bodies • mesh center equally distant from surface vertices • rotating bodies
Bounding sphere generation Live Demo
Collision detection • Minimum bounding sphere • Center – the center of the segment, connecting the two farthest mesh vertices • Radius – the half-length of the segment, connecting the two farthest mesh vertices • Efficient with • convex, oval bodies • rotating bodies • Sphere center rotated with the other mesh vertices
Minimum bounding sphere generation Live Demo
Collision detection • Axis-aligned bounding box (AABB) • Very fast to check for collisions • Usually smaller volume than bounding spheres • Edges parallel to coordinate axes • Minimum corner • coordinates – lowest coordinate ends of mesh • Maximum corner • coordinates – highest coordinate ends of mesh
Collision detection • Axis-aligned bounding box (2) • Efficient with • non-rotating bodies • convex bodies • oblong bodies • If the body rotates, the AABB needs to be recomputed
Axis-aligned bounding box generation Live Demo
Collision detection • Checking AABBs for collision • Treat the minimum and maximum corners’ coordinates as interval edges • 3D case • If the x intervals overlap • And the y intervals overlap • And the z intervals overlap • Then the AABBs intersect / collide
Collision detection • Oriented bounding box (OBB) • Generated as AABB • Rotates along with the object’s geometry • Advantage: • Rotating it is much faster than creating an new AABB • Usually less volume than AABB • Disadvantage: • Much slower collision check