问题
As per my question on Math Stackexchange:
I am working on a project for my 3D Graphics class. The project is built with C++ and OpenGL / Glut. Basically, I create a horizontal rectangle window, subdivided into two squares. On the left, I have a two dimensional coordinate plane, which allows the users to point and click and define a profile 'curve'. I then need to wrap this curve around the Y-axis n number of times.
So, would anyone be able to guide me as to how I would use Trigonometry to calculate the X and Z values of the successive points? If for example, a user clicks and creates the point:
(1, 1, 0)
And their sweep resolution (n) is set to, say, 10, then I need to redraw that point every 36 (360/10) degrees around the Y-axis.
Am I correct in assuming that Trigonometry will help me here? If so, can someone please enlighten me a bit as to how to calculate the location of a translated point in 3D space? It's been a while since I took Trig, and I don't believe we ever left 2D space.
EDIT: I attempted to use:
x'=xcos(theta)-zsin(theta)
y'=y
z'=xsin(theta)+zcos(theta)
, as per my understanding of AMPerrine's answer, and I don't think it worked as I'd hoped:
// this is in a loop
// setup the new angle
double angle = i>0 ? (360/sweepResolutionMod)*i : 0;
angle = angle * (M_PI/180);
// for each point...
for( int i=0; i<clickedPoints.size(); i++ )
{
// initial point, normalized
GLfloat tempX = (clickedPoints[i].x-250)/250;
GLfloat tempY = (clickedPoints[i].y-250)/250;
GLfloat tempZ = 0.0;
// log the initial point
cout << "(" << tempX << ", " << tempY << ", 0.0) by " << angle << " radians = ";
// generate the new point
GLfloat newX = (tempX * cos(angle)) - (tempZ * sin(angle));
GLfloat newY = tempY;
GLfloat newZ = (tempX * sin(angle)) - (tempZ * cos(angle));
// log the new point
cout << "(" << newX << ", " << newY << ", " << newZ << ")\n";
// render the new point
glVertex3d(newX, newY, newZ);
}
This produces no screen output, but console output of:
(0.048, -0.296, 0.0) by 0 radians = (0.048, -0.296, 0)
(0.376, -0.508, 0.0) by 0 radians = (0.376, -0.508, 0)
(0.72, -0.204, 0.0) by 0 radians = (0.72, -0.204, 0)
(0.652, 0.176, 0.0) by 0 radians = (0.652, 0.176, 0)
(0.368, 0.504, 0.0) by 0 radians = (0.368, 0.504, 0)
(0.048, -0.296, 0.0) by 0.628319 radians = (0.0388328, -0.296, 0.0282137)
(0.376, -0.508, 0.0) by 0.628319 radians = (0.30419, -0.508, 0.221007)
(0.72, -0.204, 0.0) by 0.628319 radians = (0.582492, -0.204, 0.423205)
(0.652, 0.176, 0.0) by 0.628319 radians = (0.527479, 0.176, 0.383236)
(0.368, 0.504, 0.0) by 0.628319 radians = (0.297718, 0.504, 0.216305)
(0.048, -0.296, 0.0) by 1.25664 radians = (0.0148328, -0.296, 0.0456507)
(0.376, -0.508, 0.0) by 1.25664 radians = (0.11619, -0.508, 0.357597)
(0.72, -0.204, 0.0) by 1.25664 radians = (0.222492, -0.204, 0.684761)
(0.652, 0.176, 0.0) by 1.25664 radians = (0.201479, 0.176, 0.620089)
(0.368, 0.504, 0.0) by 1.25664 radians = (0.113718, 0.504, 0.349989)
...
(0.048, -0.296, 0.0) by 6.28319 radians = (0.048, -0.296, -1.17566e-17)
(0.376, -0.508, 0.0) by 6.28319 radians = (0.376, -0.508, -9.20934e-17)
(0.72, -0.204, 0.0) by 6.28319 radians = (0.72, -0.204, -1.76349e-16)
(0.652, 0.176, 0.0) by 6.28319 radians = (0.652, 0.176, -1.59694e-16)
(0.368, 0.504, 0.0) by 6.28319 radians = (0.368, 0.504, -9.0134e-17)
I'm not sure what exactly is going on here, but I'm having a terrible time trying to figure it out, so please don't think I'm trying to get double reputation or anything, I'm just really stuck.
EDIT 2: Here is my whole display routine for my perspective subview:
void displayPersp(void)
{
glClear(GL_COLOR_BUFFER_BIT);
glMatrixMode (GL_MODELVIEW);
glLoadIdentity ();
gluLookAt (-2.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0);
// draw the axis
glBegin(GL_LINES);
// x
glVertex3f(500.0, 0.0, 0.0);
glVertex3f(-500.0, 0.0, 0.0);
// y
glVertex3f(0.0, -500.0, 0.0);
glVertex3f(0.0, 500.0, 0.0);
// z
glVertex3f(0.0, 0.0, -500.0);
glVertex3f(0.0, 0.0, 500.0);
glEnd();
cout << endl;
// loop as many number of times as we are going to draw the points around the Y-Axis
for( int i=0; i<=sweepResolutionMod; i++ )
{
cout << endl;
// setup the new angle
double angle = i>0 ? (360/sweepResolutionMod)*i : 0;
angle = angle * (M_PI/180);
// for each point...
for( int i=0; i<clickedPoints.size(); i++ )
{
GLfloat tempX = (clickedPoints[i].x-250)/250;
GLfloat tempY = (clickedPoints[i].y-250)/250;
GLfloat tempZ = 0.0;
cout << "(" << tempX << ", " << tempY << ", 0.0) by " << angle << " degrees = ";
GLfloat newX = (tempX * cos(angle)) - (tempZ * sin(angle));
GLfloat newY = tempY;
GLfloat newZ = (tempX * sin(angle)) - (tempZ * cos(angle));
cout << "(" << newX << ", " << newY << ", " << newZ << ")\n";
glVertex3d(newX, newY, newZ);
}
// the following was my old solution, using OpenGL's rotate(), but that
// didn't allow me to get back the new point's coordinates.
/*
glRotatef(angle, 0.0, 1.0, 0.0);
// draw a line?
if( clickedPoints.size() > 1 )
{
glBegin(GL_LINE_STRIP);
for(int i=0; i<clickedPoints.size(); i++ )
{
glVertex3f((clickedPoints[i].x-250)/250, (clickedPoints[i].y-250)/250, 0.0);
}
glEnd();
}
// everyone gets points
glBegin(GL_POINTS);
for(int i=0; i<clickedPoints.size(); i++ )
{
glVertex3f((clickedPoints[i].x-250)/250, (clickedPoints[i].y-250)/250, 0.0);
}
glEnd();
*/
}
glutSwapBuffers();
}
EDIT 3: Here is a terrible illustration that illustrates what I need to do. I know the perspective seems off, but what I'm attempting to acquire is the green 'horizontals' in the right subview (this is using the commented out glRotatef() code above):
FINAL EDIT (for future generations!):
Here is what I finally got working, after discussing some linear algebra with a teacher at college:
void displayPersp(void)
{
glClear(GL_COLOR_BUFFER_BIT);
gluLookAt (-2.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, -1.0, 0.0);
glMatrixMode (GL_MODELVIEW);
glLoadIdentity ();
// draw the axis
glBegin(GL_LINES);
// x
glVertex3f(500.0, 0.0, 0.0);
glVertex3f(-500.0, 0.0, 0.0);
// y
glVertex3f(0.0, -500.0, 0.0);
glVertex3f(0.0, 500.0, 0.0);
// z
glVertex3f(0.0, 0.0, -500.0);
glVertex3f(0.0, 0.0, 500.0);
glEnd();
cout << endl;
double previousTheta = 0.0;
for( int i=0; i<=sweepResolutionMod; i++ )
{
double theta = i>0 ? (360/sweepResolutionMod)*i : 0;
theta = theta * (M_PI/180);
if( clickedPoints.size() > 1 )
{
// the 'vertical' piece
glBegin(GL_LINE_STRIP);
for(int i=0; i<clickedPoints.size(); i++ )
{
// normalize
GLfloat tempX = (clickedPoints[i].x-250)/250;
GLfloat tempY = (clickedPoints[i].y-250)/250;
GLfloat tempZ = 0.0;
// new points
GLfloat newX = ( tempX * cos(theta) ) + ( tempZ * sin(theta) );
GLfloat newY = tempY;
GLfloat newZ = ( tempZ * cos(theta) ) - ( tempX * sin(theta) );
glVertex3f(newX, newY, newZ);
}
glEnd();
// the 'horizontal' piece
if( previousTheta != theta )
{
glBegin(GL_LINES);
for(int i=0; i<clickedPoints.size(); i++ )
{
// normalize
GLfloat tempX = (clickedPoints[i].x-250)/250;
GLfloat tempY = (clickedPoints[i].y-250)/250;
GLfloat tempZ = 0.0;
// new points
GLfloat newX = ( tempX * cos(theta) ) + ( tempZ * sin(theta) );
GLfloat newY = tempY;
GLfloat newZ = ( tempZ * cos(theta) ) - ( tempX * sin(theta) );
// previous points
GLfloat previousX = ( tempX * cos(previousTheta) ) + ( tempZ * sin(previousTheta) );
GLfloat previousY = tempY;
GLfloat previousZ = ( tempZ * cos(previousTheta) ) - ( tempX * sin(previousTheta) );
// horizontal component
glVertex3f(newX, newY, newZ);
glVertex3f(previousX, previousY, previousZ);
}
glEnd();
}
}
previousTheta = theta;
}
glutSwapBuffers();
}
回答1:
Edit 2: Okay, I see the problem you're running into -- it's a limitation I'd forgotten about (so the code I'd posted previously was dead wrong and wouldn't work at all). The problem is that you're not allowed to call glRotate between a glBegin/glEnd pair -- if you do, it'll set an error flag, and no more drawing will be done.
That does mean you pretty much have to handle the rotation yourself. Fortunately, that's a bit simpler than you've tried to make it:
static const double pi = 3.1416;
for (int point=0; point<NUM_POINTS; point++) {
glBegin(GL_LINE_STRIP);
for (double theta = 0.0; theta < 2.0 * pi; theta += pi/6.0) {
double x = cos(theta);
double z = sin(theta);
glVertex3d(points[point][0]*x, points[point][1], -1.0-points[point][0]*z);
}
glEnd();
}
As-is, this code uses -1.0 along the Z axis as the center of rotation. You can obviously move that where you wish, though anything outside your clipping frustum obviously won't display.
Also note that to get a wireframe, you'll have to draw both your "vertical", and your "horizontal" lines separately, so the code will look something like this:
for (int point=0; point<NUM_POINTS; point++) {
glBegin(GL_LINE_STRIP);
for (double theta = 0.0; theta < 2.0 * pi; theta += pi/6.0) {
double x = cos(theta);
double z = sin(theta);
glVertex3d(points[point][0]*x, points[point][1], -1.0 - points[point][0]*z);
}
glEnd();
}
for (double theta = 0.0; theta < 2.0 * pi; theta += pi/6.0) {
glBegin(GL_LINE_STRIP);
for (int point=0; point<NUM_POINTS; point++) {
double x = cos(theta);
double z = sin(theta);
glVertex3d(points[point][0]*x, points[point][1], -1.0 - points[point][0]*z);
}
glEnd();
}
回答2:
The trig functions take angles in radians, not degrees.
I also suspect that your viewport isn't setup properly, which explains why you can't see anything on the screen. Typically when I think stuff isn't rendering, it usually is, however, I haven't configured the camera, lighting and other stuff correctly.
回答3:
Looks like you're trying to construct a surface of revolution/solid of revolution/"lathe object".
A working example:
#include <GL/glut.h>
#include <glm/glm.hpp>
#include <vector>
#include <cmath>
using namespace std;
using namespace glm;
struct Vertex
{
Vertex( const vec3& position, const vec3& normal )
: position( position )
, normal( normal )
{}
vec3 position;
vec3 normal;
};
// spin the pts array around the Z axis.
// pts.x will become the radius, and pts.y will become the height
// pts should be sorted by y-coordinate
vector< Vertex > Lathe( const vector< vec2 >& pts, unsigned int segments = 32 )
{
// precalculate circle points
vector< vec2 > circlePts;
for( unsigned int i = 0; i <= segments; ++i )
{
float angle = ( i / (float)segments ) * 3.14159f * 2.0f;
circlePts.push_back( vec2( cos( angle ), sin( angle ) ) );
}
// fill each layer
typedef vector< vec3 > Layer;
typedef vector< Layer > Layers;
Layers layers( pts.size(), Layer( circlePts.size() ) );
for( size_t i = 0; i < pts.size(); ++i )
{
for( unsigned int j = 0; j < circlePts.size(); ++j )
{
layers[i][j] = vec3( circlePts[j] * pts[i].x, pts[i].y );
}
}
// move through layers generating triangles
vector< Vertex > verts;
for( size_t i = 1; i < layers.size(); ++i )
{
const Layer& prvLayer = layers[ i-1 ];
const Layer& curLayer = layers[ i-0 ];
for( size_t j = 1; j < circlePts.size(); ++j )
{
// upper = cur layer
// UL -- UR
// left | 0 / | right
// = j-1 | / 1 | = j-0
// LL -- LR
// lower = prv layer
const vec3& LL = prvLayer[ j-1 ]; // lower-left
const vec3& LR = prvLayer[ j-0 ]; // lower-right
const vec3& UL = curLayer[ j-1 ]; // upper-left
const vec3& UR = curLayer[ j-0 ]; // upper-right
// triangle0: LL -> UR -> UL
const vec3 normal0 = normalize( cross( UR - LL, UL - LL ) );
verts.push_back( Vertex( LL, normal0 ) );
verts.push_back( Vertex( UR, normal0 ) );
verts.push_back( Vertex( UL, normal0 ) );
// triangle1: LL -> LR -> UR
const vec3 normal1 = normalize( cross( LR - LL, UL - LL ) );
verts.push_back( Vertex( LL, normal1 ) );
verts.push_back( Vertex( LR, normal1 ) );
verts.push_back( Vertex( UR, normal1 ) );
}
}
return verts;
}
// mouse state
int btn;
ivec2 startMouse;
ivec2 startRot, curRot;
void mouse(int button, int state, int x, int y )
{
if( button == GLUT_LEFT_BUTTON && state == GLUT_DOWN )
{
btn = button;
startMouse = ivec2( x, glutGet( GLUT_WINDOW_HEIGHT ) - y );
startRot = curRot;
}
}
void motion( int x, int y )
{
ivec2 curMouse( x, glutGet( GLUT_WINDOW_HEIGHT ) - y );
if( btn == GLUT_LEFT_BUTTON )
{
curRot = startRot + ( curMouse - startMouse );
}
glutPostRedisplay();
}
vector< Vertex > model;
void display()
{
glClear( GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT );
glMatrixMode( GL_PROJECTION );
glLoadIdentity();
double w = glutGet( GLUT_WINDOW_WIDTH );
double h = glutGet( GLUT_WINDOW_HEIGHT );
double ar = w / h;
gluPerspective( 60, ar, 0.1, 40 );
glMatrixMode( GL_MODELVIEW );
glLoadIdentity();
glTranslatef( 0, 0, -10 );
glPushMatrix();
glRotatef( curRot.x % 360, 0, 1, 0 );
glRotatef( -curRot.y % 360, 1, 0, 0 );
// draw model
if( !model.empty() )
{
glColor3ub( 255, 0, 0 );
glEnableClientState( GL_VERTEX_ARRAY );
glEnableClientState( GL_NORMAL_ARRAY );
glVertexPointer( 3, GL_FLOAT, sizeof(Vertex), &model[0].position );
glNormalPointer( GL_FLOAT, sizeof(Vertex), &model[0].normal );
glDrawArrays( GL_TRIANGLES, 0, model.size() );
glDisableClientState( GL_VERTEX_ARRAY );
glDisableClientState( GL_NORMAL_ARRAY );
}
// draw bounding cube
glDisable( GL_LIGHTING );
glColor3ub( 255, 255, 255 );
glutWireCube( 7 );
glEnable( GL_LIGHTING );
glPopMatrix();
glutSwapBuffers();
}
int main( int argc, char **argv )
{
vector< vec2 > pts;
pts.push_back( vec2( 0.1, -3 ) );
pts.push_back( vec2( 2, -2 ) );
pts.push_back( vec2( 3, -1 ) );
pts.push_back( vec2( 1, 0 ) );
pts.push_back( vec2( 3, 1 ) );
pts.push_back( vec2( 4, 2 ) );
pts.push_back( vec2( 4, 3 ) );
model = Lathe( pts );
glutInit( &argc, argv );
glutInitDisplayMode( GLUT_RGBA | GLUT_DEPTH | GLUT_DOUBLE );
glutInitWindowSize( 640, 480 );
glutCreateWindow( "GLUT" );
glutDisplayFunc( display );
glutMouseFunc( mouse );
glutMotionFunc( motion );
glEnable( GL_DEPTH_TEST );
// set up lighting
glShadeModel( GL_SMOOTH );
glEnable( GL_COLOR_MATERIAL );
glColorMaterial( GL_FRONT_AND_BACK, GL_AMBIENT_AND_DIFFUSE ) ;
glLightModeli( GL_LIGHT_MODEL_TWO_SIDE, GL_TRUE );
glEnable( GL_LIGHTING );
// set up "headlamp"-like light
glEnable( GL_LIGHT0 );
glMatrixMode( GL_MODELVIEW );
glLoadIdentity();
GLfloat position[] = { 0, 0, 1, 0 };
glLightfv( GL_LIGHT0, GL_POSITION, position );
glPolygonMode( GL_FRONT, GL_FILL );
glPolygonMode( GL_BACK, GL_LINE );
glutMainLoop();
return 0;
}
来源：https://stackoverflow.com/questions/7904281/opengl-rotate-a-curve-about-the-y-axis