GLUT Trackball Demo
eryar@163.com
1.Introduction
在三維場(chǎng)景中建立模型后，為了方便用戶從各個(gè)角度觀察模型，從而需要對(duì)三維視圖進(jìn)行控制。常見(jiàn)的視圖交互控制方式有：Trackball控制器、飛行控制器，還有三維游戲常用的第一人稱控制器。這些視圖控制器的根本是對(duì)模型視圖矩陣MODELVIEW進(jìn)行變換。
Trackball控制器以一種用戶友好的交互方式來(lái)變換視圖，原理是由Trackball激發(fā)，Trackball如下圖所示：
Figure 1. Trackball
通過(guò)手指在球面上滾動(dòng)，就可以對(duì)三維視圖進(jìn)行控制。現(xiàn)在需要用鼠標(biāo)的拖動(dòng)來(lái)模擬Trackball以實(shí)現(xiàn)對(duì)三維視圖的控制。在OpenGL中實(shí)現(xiàn)Trackball控制視圖分為以下幾步：
1.將鼠標(biāo)移動(dòng)時(shí)的屏幕坐標(biāo)點(diǎn)映射到單位球上；
2.將開(kāi)始旋轉(zhuǎn)視圖時(shí)鼠標(biāo)點(diǎn)到球心的向量與鼠標(biāo)移動(dòng)過(guò)程中的坐標(biāo)點(diǎn)球心的向量叉乘，即可得到旋轉(zhuǎn)軸；
根據(jù)叉乘的定義，可以得到旋轉(zhuǎn)角度:
有了旋轉(zhuǎn)軸和旋轉(zhuǎn)角度，就可以對(duì)視圖進(jìn)行旋轉(zhuǎn)操作了。
2.GLUT Test
為了簡(jiǎn)明地說(shuō)明Trackball的原理，這里只使用了GLUT庫(kù)和OpenCASCADE中的四元數(shù)和向量相關(guān)的類。如果其他開(kāi)源庫(kù)也有向量計(jì)算和四元數(shù)據(jù)計(jì)算類，也可以將代碼很快移植到使用其他庫(kù)，如矩陣計(jì)算庫(kù)Eigen等。下面給出GLUT的示例代碼：
/*
Copyright(C) 2017 Shing Liu(eryar@163.com)
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files(the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and / or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions :
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/
#include <gp_XYZ.hxx>
#include <gp_Trsf.hxx>
#include <gp_Quaternion.hxx>
#include <gl/glut.h>
#pragma comment(lib, "TKernel.lib")
#pragma comment(lib, "TKMath.lib")
GLint VIEWPORT_WIDTH = 0;
GLint VIEWPORT_HEIGHT = 0;
gp_XYZ U;
gp_XYZ V;
gp_Quaternion R;
gp_Quaternion Q;
void init(void)
{
GLfloat aSpecularMaterial[] = {1.0f, 1.0f, 1.0f, 1.0f};
GLfloat aLightPosition[] = {1.0, 1.0, 1.0, 0.0};
GLfloat aWhiteLight[] = {1.0, 1.0, 1.0, 1.0};
GLfloat aModelAmbient[] = {0.1, 0.1, 0.1, 1.0};
glClearColor(0.0, 0.0, 0.0, 0.0);
glShadeModel(GL_SMOOTH);
glMaterialfv(GL_FRONT, GL_SPECULAR, aSpecularMaterial);
glMaterialf(GL_FRONT, GL_SHININESS, 60.0);
glLightfv(GL_LIGHT0, GL_POSITION, aLightPosition);
glLightfv(GL_LIGHT0, GL_SPECULAR, aWhiteLight);
glLightfv(GL_LIGHT0, GL_DIFFUSE, aWhiteLight);
glLightModelfv(GL_LIGHT_MODEL_AMBIENT, aModelAmbient);
// Enable lighting
glEnable(GL_LIGHTING);
glEnable(GL_LIGHT0);
glEnable(GL_DEPTH_TEST);
}
void display(void)
{
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glutSolidTeapot(1.0);
// draw mouse motion point.
glBegin(GL_LINES);
glVertex3f(0.0, 0.0, 0.0);
glVertex3f(U.X() * 2.0, U.Y() * 2.0, U.Z() * 2.0);
glEnd();
glutSwapBuffers();
}
void reshape(GLint theWidth, GLint theHeight)
{
VIEWPORT_WIDTH = theWidth;
VIEWPORT_HEIGHT = theHeight;
// Reset viewport and projection parameter
glViewport(0, 0, theWidth, theHeight);
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
if (theWidth <= theHeight)
{
glOrtho(-1.5, 1.5, -1.5 * theHeight / theWidth, 1.5 * theHeight / theWidth, -10.0, 10.0);
}
else
{
glOrtho(-1.5 * theWidth / theHeight, 1.5 * theWidth / theHeight, -1.5, 1.5, -10.0, 10.0);
}
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
}
void mapToSphere(GLint theX, GLint theY, gp_XYZ& thePnt)
{
GLfloat aX = (theX - 0.5 * VIEWPORT_WIDTH) / VIEWPORT_WIDTH;
GLfloat aY = (0.5 * VIEWPORT_HEIGHT - theY) / VIEWPORT_HEIGHT;
GLfloat aSinx = sin(M_PI * aX * 0.5);
GLfloat aSiny = sin(M_PI * aY * 0.5);
GLfloat aSxy2 = aSinx * aSinx + aSiny * aSiny;
thePnt.SetX(aSinx);
thePnt.SetY(aSiny);
thePnt.SetZ(aSxy2 < 1.0 ? sqrt(1.0 - aSxy2) : 0.0);
}
void mouse(GLint theButton, GLint theState, GLint theX, GLint theY)
{
mapToSphere(theX, theY, U);
glutPostRedisplay();
}
void motion(GLint theX, GLint theY)
{
mapToSphere(theX, theY, V);
gp_XYZ W = U.Crossed(V);
if (W.Modulus() < gp::Resolution())
{
return;
}
GLfloat aAngle = W.Modulus() / (U.Modulus() * V.Modulus());
aAngle = asin(aAngle);
glRotatef(aAngle * 180.0 / M_PI, W.X(), W.Y(), W.Z());
glutPostRedisplay();
U = V;
}
int main(int argc, char* argv[])
{
glutInit(&argc, argv);
glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGBA);
glutInitWindowSize(500, 300);
glutCreateWindow("Trackball Demo");
init();
glutDisplayFunc(display);
glutReshapeFunc(reshape);
glutMouseFunc(mouse);
glutMotionFunc(motion);
glutMainLoop();
return 0;
}
上述程序運(yùn)行結(jié)果如下動(dòng)圖所示：
從上圖可知，當(dāng)旋轉(zhuǎn)幾次后視圖并沒(méi)有得到預(yù)期的結(jié)果。因?yàn)槌绦驅(qū)⑹髽?biāo)映射后坐標(biāo)與球心得到的向量進(jìn)行了顯示，發(fā)現(xiàn)當(dāng)旋轉(zhuǎn)幾次后，這個(gè)向量并沒(méi)有跟隨鼠標(biāo)。
3.Transform
通過(guò)觀察上面代碼程序運(yùn)行的結(jié)果，可以發(fā)現(xiàn)鼠標(biāo)映射函數(shù)得到的映射點(diǎn)始終是位于XOY平面上的一個(gè)半球面上。當(dāng)視圖被旋轉(zhuǎn)后，視圖的坐標(biāo)系已經(jīng)發(fā)生了變化，而映射點(diǎn)并沒(méi)有。為了跟蹤這個(gè)變換用四元數(shù)進(jìn)行累乘來(lái)記錄這一系列的旋轉(zhuǎn)變換。最后在映射函數(shù)中將映射點(diǎn)變換到已經(jīng)改變的視圖坐標(biāo)系中。
即在鼠標(biāo)移動(dòng)處理函數(shù)中增加記錄變換：
gp_Quaternion q(W, aAngle);
R.Multiply(q);
在mapToSphere函數(shù)中增加：
gp_Trsf aTrsf;
aTrsf.SetRotation(Q.Inverted());
aTrsf.Transforms(thePnt);
列出升級(jí)后的全部代碼如下所示：
/*
Copyright(C) 2017 Shing Liu(eryar@163.com)
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files(the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and / or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions :
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/
#include <gp_XYZ.hxx>
#include <gp_Trsf.hxx>
#include <gp_Quaternion.hxx>
#include <gl/glut.h>
#pragma comment(lib, "TKernel.lib")
#pragma comment(lib, "TKMath.lib")
GLint VIEWPORT_WIDTH = 0;
GLint VIEWPORT_HEIGHT = 0;
gp_XYZ U;
gp_XYZ V;
gp_Quaternion R;
gp_Quaternion Q;
void init(void)
{
GLfloat aSpecularMaterial[] = {1.0f, 1.0f, 1.0f, 1.0f};
GLfloat aLightPosition[] = {1.0, 1.0, 1.0, 0.0};
GLfloat aWhiteLight[] = {1.0, 1.0, 1.0, 1.0};
GLfloat aModelAmbient[] = {0.1, 0.1, 0.1, 1.0};
glClearColor(0.0, 0.0, 0.0, 0.0);
glShadeModel(GL_SMOOTH);
glMaterialfv(GL_FRONT, GL_SPECULAR, aSpecularMaterial);
glMaterialf(GL_FRONT, GL_SHININESS, 60.0);
glLightfv(GL_LIGHT0, GL_POSITION, aLightPosition);
glLightfv(GL_LIGHT0, GL_SPECULAR, aWhiteLight);
glLightfv(GL_LIGHT0, GL_DIFFUSE, aWhiteLight);
glLightModelfv(GL_LIGHT_MODEL_AMBIENT, aModelAmbient);
// Enable lighting
glEnable(GL_LIGHTING);
glEnable(GL_LIGHT0);
glEnable(GL_DEPTH_TEST);
}
void display(void)
{
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glutSolidTeapot(1.0);
// draw mouse motion point.
glBegin(GL_LINES);
glVertex3f(0.0, 0.0, 0.0);
glVertex3f(U.X() * 2.0, U.Y() * 2.0, U.Z() * 2.0);
glEnd();
glutSwapBuffers();
}
void reshape(GLint theWidth, GLint theHeight)
{
VIEWPORT_WIDTH = theWidth;
VIEWPORT_HEIGHT = theHeight;
// Reset viewport and projection parameter
glViewport(0, 0, theWidth, theHeight);
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
if (theWidth <= theHeight)
{
glOrtho(-1.5, 1.5, -1.5 * theHeight / theWidth, 1.5 * theHeight / theWidth, -10.0, 10.0);
}
else
{
glOrtho(-1.5 * theWidth / theHeight, 1.5 * theWidth / theHeight, -1.5, 1.5, -10.0, 10.0);
}
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
}
void mapToSphere(GLint theX, GLint theY, gp_XYZ& thePnt)
{
GLfloat aX = (theX - 0.5 * VIEWPORT_WIDTH) / VIEWPORT_WIDTH;
GLfloat aY = (0.5 * VIEWPORT_HEIGHT - theY) / VIEWPORT_HEIGHT;
GLfloat aSinx = sin(M_PI * aX * 0.5);
GLfloat aSiny = sin(M_PI * aY * 0.5);
GLfloat aSxy2 = aSinx * aSinx + aSiny * aSiny;
thePnt.SetX(aSinx);
thePnt.SetY(aSiny);
thePnt.SetZ(aSxy2 < 1.0 ? sqrt(1.0 - aSxy2) : 0.0);
gp_Trsf aTrsf;
aTrsf.SetRotation(Q.Inverted());
aTrsf.Transforms(thePnt);
}
void mouse(GLint theButton, GLint theState, GLint theX, GLint theY)
{
mapToSphere(theX, theY, U);
Q = R;
glutPostRedisplay();
}
void motion(GLint theX, GLint theY)
{
mapToSphere(theX, theY, V);
gp_XYZ W = U.Crossed(V);
if (W.Modulus() < gp::Resolution())
{
return;
}
GLfloat aAngle = W.Modulus() / (U.Modulus() * V.Modulus());
aAngle = asin(aAngle);
glRotatef(aAngle * 180.0 / M_PI, W.X(), W.Y(), W.Z());
glutPostRedisplay();
gp_Quaternion q(W, aAngle);
R.Multiply(q);
U = V;
}
int main(int argc, char* argv[])
{
glutInit(&argc, argv);
glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGBA);
glutInitWindowSize(500, 300);
glutCreateWindow("Trackball Demo");
init();
glutDisplayFunc(display);
glutReshapeFunc(reshape);
glutMouseFunc(mouse);
glutMotionFunc(motion);
glutMainLoop();
return 0;
}
這次程序運(yùn)行和預(yù)期結(jié)果一致，旋轉(zhuǎn)很流暢：
4.Conclusion
程序員總是有很強(qiáng)的控制欲，希望一切盡在掌握之中。在三維場(chǎng)景中建立模型后，如何對(duì)視圖進(jìn)行控制來(lái)方便地觀察模型呢？最常見(jiàn)的控制方式就是Trackball. OpenSceneGraph、Eigen等開(kāi)源庫(kù)都有相關(guān)的實(shí)現(xiàn)。
Trackball的實(shí)現(xiàn)主要是將鼠標(biāo)點(diǎn)映射到一個(gè)球面上，然后使用叉乘得到旋轉(zhuǎn)軸和旋轉(zhuǎn)角度。為了旋轉(zhuǎn)的流暢，使用四元數(shù)記錄了一系列的旋轉(zhuǎn)變換，最后通過(guò)將映射點(diǎn)進(jìn)行坐標(biāo)變換得到滿意的效果。
5.References
1. Virtual Trackball. http://gukewen.sdu.edu.cn/panrj/courses/4-AngelCGE2-Virtual-Trackball.pdf
2. Object Mouse Trackball https://www.khronos.org/opengl/wiki/Object_Mouse_Trackball