game programming 05 3d math for games

Game Programming 053D math for Games

2010 년 2 학기디지털콘텐츠전공

Unbuffered Input

Call-back Driven Framework

• Framework:– A partially-constructed application

• Encapsulating knowledge of how to use the low-level libraries

• Providing functions for the game flow • But functions are mostly empty

– Need to be completed by a programmer• Fill-in the missing details• Overide call-back functions


• Example from Ogre3D’s game loop

while (true){

for (each frameListener){

frameListener.frameStarted();}

renderCurrentScene();


frameListener.frameEnded();}

}

while (true){


frameListener.frameStarted();}

renderCurrentScene();


frameListener.frameEnded();}

}


• Using Ogre’s FrameListenerClass GameFrameListener : public Ogre::FrameListener{public:

virtual void frameStarted(const FrameEvent &event){

// Do things happening before renderingpollJoypad(event);updatePlayerControls(events);updateDynamicSimulation(events);resolveCollisions(events);updateCamera(events);// etc.

}virtual void frameEnded(const FrameEvent &event){

// Do things after renderingdrawHud(events);

}}

Class GameFrameListener : public Ogre::FrameListener{public:

virtual void frameStarted(const FrameEvent &event){

// Do things happening before renderingpollJoypad(event);updatePlayerControls(events);updateDynamicSimulation(events);resolveCollisions(events);updateCamera(events);// etc.

}virtual void frameEnded(const FrameEvent &event){

// Do things after renderingdrawHud(events);

}}

OGRE3D: Frame Listener

• 한 장면이 화면에 렌더링되기 직전 /직후에 해야 할 일을 정의하고 있는 객체

• 추가 작업을 위해서는 frame listener 를 추가 생성하고 , 이를 root 에 등록시켜 사용

• 여러 개의 frame listener 가 존재할 수 있다 .

Frame Listener 의 맴버함수

frameStarted Called just before a frame is rendered.

frameRenderingQueued

Called after all render targets have had their rendering commands issued, but beforethe render windows have been asked to flip their buffers over

frameEnded Called just after a frame has been rendered.

virtual bool frameStarted(const FrameEvent& evt); virtual bool frameRenderingQueued(const FrameEvent& evt); virtual bool frameEnded(const FrameEvent& evt)

virtual bool frameStarted(const FrameEvent& evt); virtual bool frameRenderingQueued(const FrameEvent& evt); virtual bool frameEnded(const FrameEvent& evt)

FrameEvent structure: Real timeSinceLastEvent, Real timeSinceLastFrame 값을 기억하는 구조체

Frame Listener 의 호출• Root 객체의 rendering loop 내에서 호출

bool Root::renderOneFrame(void){ if(!_fireFrameStarted()) return false; if (!_updateAllRenderTargets()) return false; return _fireFrameEnded();}

bool Root::renderOneFrame(void){ if(!_fireFrameStarted()) return false; if (!_updateAllRenderTargets()) return false; return _fireFrameEnded();}

Ogre Engine 의 Rendering Loop• Root::startRendering() 함수에서 이뤄짐• Redering Loop 실행

– 매 프레임 마다 renderOneFrame 실행 :

• Loop 의 중단– FrameStarted, frameRenderingQueued, FrameEnded 에서

하나라도 false 를 return 할 경우

1. Frame Listener 들의 FrameStarted() 함수 호출1. Frame Listener 들의 FrameStarted() 함수 호출

3. Frame Listener 들의 FrameEnded() 함수 호출3. Frame Listener 들의 FrameEnded() 함수 호출

Frame Listener 들의 frameRenderingQueued() 함수 호출Frame Listener 들의 frameRenderingQueued() 함수 호출

2. 한 프레임 렌더링2. 한 프레임 렌더링

Input Handling

• Types of input handling– Unbuffered Input

• Testing states of Mouse/Keyboard every frame• Do action if necessary

– Buffered Input• Event(message)-driven• Event is queued into a buffer• Call message pump routines per frame

Ninja 움직이기• Add two functions in Application class:

class OgreApp : public BaseApplication{public: BasicTutorial4(void); virtual ~BasicTutorial4(void);protected: virtual void createScene(void); virtual bool frameRenderingQueued(const Ogre::FrameEvent& evt);private: bool processUnbufferedInput(const Ogre::FrameEvent& evt);}

class OgreApp : public BaseApplication{public: BasicTutorial4(void); virtual ~BasicTutorial4(void);protected: virtual void createScene(void); virtual bool frameRenderingQueued(const Ogre::FrameEvent& evt);private: bool processUnbufferedInput(const Ogre::FrameEvent& evt);}

CreateScene 수정• Ninja 를 포함하고 light 를 세팅

mSceneMgr->setAmbientLight(Ogre::ColourValue(0.25, 0.25, 0.25));

Ogre::Entity* ninjaEntity = mSceneMgr->createEntity("Ninja", "ninja.mesh");Ogre::SceneNode *node = mSceneMgr->getRootSceneNode()

->createChildSceneNode("NinjaNode");node->attachObject(ninjaEntity);

Ogre::Light* pointLight = mSceneMgr->createLight("pointLight");pointLight->setType(Ogre::Light::LT_POINT);pointLight->setPosition(Ogre::Vector3(250, 150, 250));pointLight->setDiffuseColour(Ogre::ColourValue::White);pointLight->setSpecularColour(Ogre::ColourValue::White);

mSceneMgr->setAmbientLight(Ogre::ColourValue(0.25, 0.25, 0.25));

Ogre::Entity* ninjaEntity = mSceneMgr->createEntity("Ninja", "ninja.mesh");Ogre::SceneNode *node = mSceneMgr->getRootSceneNode()

->createChildSceneNode("NinjaNode");node->attachObject(ninjaEntity);

Ogre::Light* pointLight = mSceneMgr->createLight("pointLight");pointLight->setType(Ogre::Light::LT_POINT);pointLight->setPosition(Ogre::Vector3(250, 150, 250));pointLight->setDiffuseColour(Ogre::ColourValue::White);pointLight->setSpecularColour(Ogre::ColourValue::White);

Frame Listner 수정• frameRenderingQueued 함수 override

bool OgreApp::frameRenderingQueued(const Ogre::FrameEvent& evt){ bool ret = BaseApplication::frameRenderingQueued(evt); if(!processUnbufferedInput(evt)) return false; return ret;}

bool OgreApp::frameRenderingQueued(const Ogre::FrameEvent& evt){ bool ret = BaseApplication::frameRenderingQueued(evt); if(!processUnbufferedInput(evt)) return false; return ret;}

Processing Input

bool OgreApp::processUnbufferedInput(const Ogre::FrameEvent& evt){ static bool mMouseDown = false; // If a mouse button is depressed static Ogre::Real mToggle = 0.0; // The time left until next toggle static Ogre::Real mRotate = 0.13; // The rotate constant static Ogre::Real mMove = 250; // The movement constant


Processing Input

• UnbufferedInput 구현– Mouse 와 keyboard 의 현재 상태를 얻어옴

(BaseApplication::frameRenderingQueued 에서 수행 )

• 다음과 같이 사용할 준비를 한다 . bool OgreApp::processUnbufferedInput(const Ogre::FrameEvent& evt){ static bool mMouseDown = false; // If a mouse button is depressed static Ogre::Real mToggle = 0.0; // The time left until next toggle static Ogre::Real mRotate = 0.13; // The rotate constant static Ogre::Real mMove = 250; // The movement constant


mMouse->capture(); mKeyboard->capture();mMouse->capture(); mKeyboard->capture();

Toggle 기능 구현• 현재 mouse 와 마지막 mouse 상태의 비교를

통해 구현

bool currMouse = mMouse->getMouseState().buttonDown(OIS::MB_Left);

if (currMouse && ! mMouseDown){ Ogre::Light* light = mSceneMgr->getLight("pointLight"); light->setVisible(! light->isVisible());}

mMouseDown = currMouse;

bool currMouse = mMouse->getMouseState().buttonDown(OIS::MB_Left);

if (currMouse && ! mMouseDown){ Ogre::Light* light = mSceneMgr->getLight("pointLight"); light->setVisible(! light->isVisible());}

mMouseDown = currMouse;

Delayed Toggle 구현• 눌린 시간을 기억하여 어느 일정시간이상

눌렸으면 실행

mToggle -= evt.timeSinceLastFrame;

if ((mToggle < 0.0f ) && mKeyboard->isKeyDown(OIS::KC_1)){ mToggle = 0.5; Ogre::Light* light = mSceneMgr->getLight("pointLight"); light->setVisible(! light->isVisible());}

mToggle -= evt.timeSinceLastFrame;

if ((mToggle < 0.0f ) && mKeyboard->isKeyDown(OIS::KC_1)){ mToggle = 0.5; Ogre::Light* light = mSceneMgr->getLight("pointLight"); light->setVisible(! light->isVisible());}

Keyboard 입력 구현• 키보드의 상태를 찾아내어 구현

Ogre::Vector3 transVector = Ogre::Vector3::ZERO;

if (mKeyboard->isKeyDown(OIS::KC_I)) // ForwardtransVector.z -= mMove;

if (mKeyboard->isKeyDown(OIS::KC_K)) // BackwardtransVector.z += mMove;

if (mKeyboard->isKeyDown(OIS::KC_U)) // UptransVector.y += mMove;

if (mKeyboard->isKeyDown(OIS::KC_O)) // DowntransVector.y -= mMove;

mSceneMgr->getSceneNode("NinjaNode")->translate(transVector * evt.timeSinceLastFrame, Ogre::Node::TS_LOCAL);







Rotate 구현• SHIFT 키와 조합하여 구현

if (mKeyboard->isKeyDown(OIS::KC_J)) // Left - yaw or strafe{ if(mKeyboard->isKeyDown( OIS::KC_LSHIFT )) { // Yaw left mSceneMgr->getSceneNode("NinjaNode")->yaw(Ogre::Degree(mRotate * 5)); } else { transVector.x -= mMove; // Strafe left }}if (mKeyboard->isKeyDown(OIS::KC_L)) // Right - yaw or strafe{ if(mKeyboard->isKeyDown( OIS::KC_LSHIFT )) { // Yaw right mSceneMgr->getSceneNode("NinjaNode")->yaw(Ogre::Degree(-mRotate * 5)); } else { transVector.x += mMove; // Strafe right }}

if (mKeyboard->isKeyDown(OIS::KC_J)) // Left - yaw or strafe{ if(mKeyboard->isKeyDown( OIS::KC_LSHIFT )) { // Yaw left mSceneMgr->getSceneNode("NinjaNode")->yaw(Ogre::Degree(mRotate * 5)); } else { transVector.x -= mMove; // Strafe left }}if (mKeyboard->isKeyDown(OIS::KC_L)) // Right - yaw or strafe{ if(mKeyboard->isKeyDown( OIS::KC_LSHIFT )) { // Yaw right mSceneMgr->getSceneNode("NinjaNode")->yaw(Ogre::Degree(-mRotate * 5)); } else { transVector.x += mMove; // Strafe right }}

Buffered Input

• Unbuffered Input: – Every frame, the state of

OIS::Keyboard/OIS::Mouse is queried

• Buffered Input– Event-driven– Event:

• key-pressed/released• Mouse button pressed/released• Mouse move

Event-Driven Updating

• Event:– Any interesting change in the state of the

game– Examples:

• The player pressing a keyboard• The explosion going off• An enemy character spotting the player

Event-Driven Updating

• Event System– Similar to windows GUI– A subsystem of a game engine

• Register events of interests• Respond to the events when they occur

– Periodic services can be implemented by an event

• Example: Posting a event for every 1/30 sec.• Using the event que for posting a event for the future

Keyboard 입력 구현• 키보드의 상태를 찾아내어 구현













3D Math for Games

• A game is:– a mathematical model of a virtual world

simulated in real-time on a computer.

• So, Mathematics provides everything !

• Games programmers must use of:– All branches of mathematics:

Trigonometry, algebra, statistics, calculus, and so on

Points

• Point: a location in n-dimensional space

• Coordinate systems:– Cartesian coordinate

– Cylindrical coordinate

– Spherical coordinate

𝐩= (𝑝1,𝑝2,𝑝3)

Vectors

• Point: absolute• Vector: relative (direction and magnitude)

• Basic Operations:– Multiplication by a scalar

– Addition and Subtraction

– Magnitude (length, norm)

(in a cartesian coordinate system)𝐚= (𝑎1,𝑎2,𝑎3)

Vectors

• Normalization and Unit vector

• Normal Vector

Vectors

• Dot product

– Dot product in action:• Collinear test• Collinear but opposite • Perpendicular test• Same direction (same side)• Opposite direction (opposite side)• Compute height from a plane

Vectors

• Cross product

– Cross product in action:• Find perpendicular direction from the given two

vectors• Compute torque (rotational force)

Vectors in Action!

• Ogre::Vector3 ClassOgreVector3.hOgreVector3.h

Linear Interpolation

• LERP (linear interpolation)– Given two points (vectors), finding inbetweens

P

Qv = Q - P

P + α v= P + α (Q – P)= (1- α) P + α Q

α

Matrices

• Basic Operations:– Addition

– Scalar multiplication

– Transpose

Matrices

• Matrix Multiplication

• Identity Matrix

AB ≠ BA

Matrices

• Vectors as a Matrix– Row vector

– Column vector

aM=b

Mp=q

=

Homogeneous Coordinates

• Any affine transformation between 3D spaces can be represented by a 4x4 matrix

110)( 131333 pTMpT

Transformations

• 2D rotation

• 2D scaling

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Transformations

• 2D shear

• 2D reflection

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Affine Transformations

• 2D translation

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Examples of Affine Transformations• 2D transformation for vectors

– Translation is simply ignored

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Affine Transformations

• 3D rotation

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Pivot-Point Rotation• Rotation with respect to a pivot point (x,y)

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sin)cos1(sincos

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Fixed-Point Scaling• Scaling with respect to a fixed point (x,y)

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Scaling Direction

• Scaling along an arbitrary axis

)(),()(1 RssSR yx

)(1 R),( yx ssS)(R

Matrices in Action

• Ogre::Matrix3, Ogre::Matrix4

Euler angles

• Arbitrary rotation can be represented by three rotation along x,y,z axis

1000

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)()()(),,(

CCSCS

SCCSSCCSSSCS

SSCSC

RRRR xyzXYZ

Gimble

• Hardware implementation of Euler angles• Aircraft, Camera

Euler Angles

• Rotation about three orthogonal axes– 12 combinations

• XYZ, XYX, XZY, XZX• YZX, YZY, YXZ, YXY• ZXY, ZXZ, ZYX, ZYZ

• Gimble lock– Coincidence of inner most

and outmost gimbles’ rotation axes

– Loss of degree of freedom

Euler Angles

• Euler angles are ambiguous– Two different Euler angles can represent the

same orientation

– This ambiguity brings unexpected results of animation where frames are generated by interpolation.

),2

,0 and )0,2

, 21 (R(θ),r,r(rR zyx

Rotation and orientation

• Euler theorem– Every 3D frame can be written as a spinning by

θ about a fixed axis n, the eigenvector of the rotation matrix.

Problems with Euler angles

• Gimbal lock– Losing a degree of freedom

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z

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1 1

2

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• How to Interpolate orientations with Euler angles?

y

z

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• What we want is– A parameterization of orientation enabling us

to• Resolve the gimbal lock problem.• Get a simple steady rotation between any two

orientation. (Euler theorem)• Define rotational moves that are independent of the

choice of the coordinate system.

• Such a parameterization exists?– Yes, Unit quaternions.

Quaternion

• A complex number of four numbers

• Unit quaternion– A quaternion of unit length

– A point on the unit hyper-sphere S3

),,,( zyxwq

12222 zyxw

Representation

• Grouping the four components into a scalar and a 3D vector

),(),,,( vq wzyxw

Quaternion and Rotation

• Representing an rotation by a unit quaternion

)2

sinˆ,2

(cos

nq

Antipodal Points in S3

• and are the same orientationq q

qnnq

))2

2sin(ˆ),

2

2(cos( ),

2sinˆ,

2(cos

Algebra of quaternions

• Addition

• Multiplication– Non-commutative under multiplication

)](),[(

),(),(

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),(),(),( 1221111121 vvvvqq wwww


• Inverse operation

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• Rotation with a unit quaternion q

• Successive rotation with quaternions

,)( 1qpqpRq

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)()()()())((1212

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Into and Out of Quaternion Space• unit quaternion to rotation matrix

,

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02222122

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22

22

22

yxwxyzwyxz

wxyzzxwzxy

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qR

).,(),,,( where vq wzyxw

Into and Out of Quaternion Space• Rotation matrix to unit quaternion

,

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Quaternions in Action

• Ogre::QuaternionOgreQuaternion.hOgreQuaternion.h

game programming 05 3d math for games

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