This program demonstrates the creation of display lists for constructing some quadric surfaces. More...

#include <IL/il.h>
#include <GL/glut.h>
#include <stdio.h>
#include <math.h>
#include <stdlib.h>
#include <string.h>
#include <sys/time.h>
#include <Arcball/Arcball.hpp>

Include dependency graph for torus.cpp:

Go to the source code of this file.

Data Structures
struct	_Point
	3D point structure. More...

Macros
#define	GL_GLEXT_PROTOTYPES

#define	M_PI_ 3.14159265358979323846
	Π definition.

#define	checkImageWidth 128
	Texture Image width.

#define	checkImageHeight 128
	Texture Image height.

#define	farPlane (zobs+10*d)
	Far plane distance.

#define	frontPlane 1
	Front plane distance.

#define	posAng(x) ((x) < 0.0 ? ((x)+TWOPI) : (x))
	Return a positive angle.

#define	TWOPI (2*M_PI)
	Two PI.

#define	toRad(x) ((x)*M_PI/180.0)
	Convert an angle to radians.

#define	toDeg(x) ((x)*180.0/M_PI)
	Convert an angle to degrees.

#define	clamp(x) (fmin(fmax((x),0),1))
	Clamp to [0,1].

#define	LEN(x, y, z) (sqrt((x)(x)+(y)(y)+(z)*(z)))
	Vector length.

#define	theAxes (2*nObjects)
	Axes.

#define	nObjects (sizeof(objNames)/sizeof(char*))
	Number of objects;.

#define	scale (0.05*diag)
	Scale for drawing normals.

Typedefs
typedef struct _Point	Point
	3D point structure.

typedef Point	_BBOX[2]
	Axis aligned Bounding box.

typedef enum _oType	oType
	Quadric types.

typedef enum _rAxes	rAxes
	Rotation axes.

Enumerations
enum	_oType { _torus = 0 , _cylinder , _cone , _ellipsoid , _paraboloid , _hyperbolic_paraboloid , _hyperboloid }
	Quadric types. More...

enum	_rAxes { xAxis , yAxis , zAxis }
	Rotation axes. More...

Functions
void	updateBox (oType index, double x, double y, double z)
	Add a new point to a bounding box.

double	getBoxSize (oType index, double dx, double dy, double *dz)
	Return bounding box dimension.

void	cone (double h, double r, int vs, int rs, int normals)
	Draw a cone, centered at the origin, with a circular base on plane XY.

void	cylinder (double h, double r, int vs, int rs, int normals)
	Draw a cylinder, centered at the origin, with a circular base on plane XY.

void	paraboloid (double h, double r, int vs, int rs, int normals)
	Draw a paraboloid, centered at the origin, with a circular base on plane XY.

void	hyperboloid (double h, double a, double r, int vs, int rs, int normals)
	Draw a one-sheeted hyperboloid of revolution, centered at the origin, with a circular base parallel to plane XY.

void	hyperbolic_paraboloid (double w, double h, int ws, int hs, int normals)
	Draw a hyperbolic paraboloid.

void	torus (double r1, double r2, int numc, int numt, int normals)
	Draw a torus, centered at the origin, and azimuthally symmetric about the z-axis.

void	sphere (double r1, double r2, double r3, int numc, int numt, int normals)
	Draw an ellipsoid or a sphere centered at the origin.

void	drawAxes (double r1, double r2)
	Draw the three coordinate axes.

void	drawBox (const _BBOX BBOX)
	Draw the bounding box as a dashed hexahedron.

void	makeCheckImage (void)
	Create checkerboard texture.

int	loadImage (const char *filename)
	Load an image from a file.

void	initTexture (void)
	Initialize texture stuff.

void	init (void)
	Initialize OpenGL state.

void	display (void)
	Clear window, reset depth buffer and draw torus.

int	dateNow ()
	Return the current Coordinated Universal Time (UTC) in miliseconds.

void	_timer (int value)
	Timer callback function to be triggered in at least delay milliseconds.

void	reshape (int w, int h)
	Handle window resize.

void	keyboard (unsigned char key, int x, int y)
	Keyboard callback interface.

void	mouseFunc (int state, int button, int _x, int _y)
	Mouse callback interface.

void	mouseDrag (int _x, int _y)
	Mouse callback interface.

int	main (int argc, char **argv)
	Main program for testing.

Variables
double	zobs = 10.0
	Camera position.

double	viewingAngle = 45
	Camera viewing angle, also known as opening angle.

GLuint	objects
	Object display list base index.

double	R1 = 1.0
	Distance from the center of the tube to the center of the torus.

double	R2 = 0.3
	Radius of the tube.

double	R3 = 0.5
	Radius in z direction for an ellipsoid.

int	nR1 = 25
	Number of samples.

int	nR2 = 10
	Number of samples.

int	drawWire = 0
	Whether draw wireframe or filled polygons.

int	drawNormals = 0
	Whether display vertex normals.

int	applyTexture = 0
	Whether to apply texture.

int	showAxes = 0
	Whether to show the coordinate axes.

int	showBox = 0
	Whether to show the bounding boxes.

int	backG = 0
	Toggle background color.

int	procImage = 1
	Whether to use a procedural texture.

int	useArcball = 0
	Whether to use the arcball paradigm.

int	spin = 1
	Automatic spin.

rAxes	axis = xAxis
	Current axis.

GLfloat	rotAngle = 2.0
	Rotation angle increment.

unsigned int	delay = 60
	Controls the trigger of the timer function.

GLubyte	checkImage [checkImageHeight][checkImageWidth][4]
	Procedural image for a checkerboard.

GLuint	texName
	Texture identifier.

ILuint	ilImage
	IL image identifier.

const char *	objNames [] = {"Torus", "Cylinder", "Cone", "Ellipsoid", "Paraboloid", "Hyperbolic Paraboloid", "Hyperboloid"}
	Object names.

_BBOX	objBoxes [nObjects]
	Object boxes.

int	OBJECT = _torus
	Current object.

Arcball	arcball
	ArcBall.

Detailed Description

This program demonstrates the creation of display lists for constructing some quadric surfaces.

Quadric surfaces are defined by quadratic equations in two dimensional space.

Display list is a group of OpenGL commands that have been stored (compiled) for later execution. Once a display list is created, all vertex and pixel data are evaluated and copied into the display list memory on the server machine. It is only one time process. After the display list has been prepared (compiled), you can reuse it repeatedly without re-evaluating and re-transmitting data over and over again to draw each frame. Display list is one of the fastest methods to draw static data because vertex data and OpenGL commands are cached in the display list and minimize data transmissions from the client to the server side. It means that it reduces CPU cycles to perform the actual data transfer.

To Compile:
- gcc torus.cpp -o torus-linux64
or, for MacOS:
- gcc torus.cpp -o torus.osx

Usage:

   - torus.osx R   r   r'  nR  nr
   - torus.osx 1.0 0.3 0.5 25  10

Requirements:
- DevIL-ILUT-devel (Fedora)
- libdevil1c2 libdevil-dev (Ubuntu)
- libdevil (MacPorts)

Author: Paulo R Cavalcanti

Date: 06/11/2017

See also: https://www.opengl.org/archives/resources/code/samples/redbook/torus.c; http://www.songho.ca/opengl/gl_displaylist.html; https://web.cs.wpi.edu/~matt/courses/cs563/talks/renderman/quadric.html; http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4037402; http://www.nptel.ac.in/courses/Webcourse-contents/IIT-Delhi/Computer%20Aided%20Design%20&%20ManufacturingI/mod3/01.htm; http://openil.sourceforge.net/; http://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx; http://www.cs.colby.edu/maxwell/courses/tutorials/maketutor/

Definition in file torus.cpp.

Macro Definition Documentation

◆ checkImageHeight

#define checkImageHeight 128

Texture Image height.

Definition at line 63 of file torus.cpp.

◆ checkImageWidth

#define checkImageWidth 128

Texture Image width.

Definition at line 61 of file torus.cpp.

◆ clamp

#define clamp ( x ) (fmin(fmax((x),0),1))

Clamp to [0,1].

Definition at line 79 of file torus.cpp.

◆ farPlane

#define farPlane (zobs+10*d)

Far plane distance.

Near and far clipping planes should always be specified as positive. That's because they represent how far in front of the "camera" the clipping planes will be.

Definition at line 67 of file torus.cpp.

◆ frontPlane

#define frontPlane 1

Front plane distance.

Definition at line 69 of file torus.cpp.

◆ GL_GLEXT_PROTOTYPES

#define GL_GLEXT_PROTOTYPES

Definition at line 44 of file torus.cpp.

◆ LEN

#define LEN	(	x,
		y,
		z
	)	(sqrt((x)(x)+(y)(y)+(z)*(z)))

Vector length.

Definition at line 81 of file torus.cpp.

◆ M_PI_

#define M_PI_ 3.14159265358979323846

Π definition.

Unused. We are using M_PI from math.h The difference between const and define is that const has a type and define is trick, because it is the pre-compiler which replaces the value.

Definition at line 59 of file torus.cpp.

◆ nObjects

#define nObjects (sizeof(objNames)/sizeof(char*))

Number of objects;.

Definition at line 85 of file torus.cpp.

◆ posAng

#define posAng ( x ) ((x) < 0.0 ? ((x)+TWOPI) : (x))

Return a positive angle.

Definition at line 71 of file torus.cpp.

◆ scale

#define scale (0.05*diag)

Scale for drawing normals.

Definition at line 87 of file torus.cpp.

◆ theAxes

#define theAxes (2*nObjects)

Axes.

Definition at line 83 of file torus.cpp.

◆ toDeg

#define toDeg ( x ) ((x)*180.0/M_PI)

Convert an angle to degrees.

Definition at line 77 of file torus.cpp.

◆ toRad

#define toRad ( x ) ((x)*M_PI/180.0)

Convert an angle to radians.

Definition at line 75 of file torus.cpp.

◆ TWOPI

#define TWOPI (2*M_PI)

Two PI.

Definition at line 73 of file torus.cpp.

Typedef Documentation

◆ _BBOX

typedef Point _BBOX[2]

Axis aligned Bounding box.

Definition at line 128 of file torus.cpp.

◆ oType

typedef enum _oType oType

Quadric types.

◆ Point

typedef struct _Point Point

3D point structure.

◆ rAxes

typedef enum _rAxes rAxes

Rotation axes.

Enumeration Type Documentation

◆ _oType

enum _oType

Quadric types.

Enumerator
_torus
_cylinder
_cone
_ellipsoid
_paraboloid
_hyperbolic_paraboloid
_hyperboloid

Definition at line 131 of file torus.cpp.

◆ _rAxes

enum _rAxes

Rotation axes.

Enumerator
xAxis
yAxis
zAxis

Definition at line 134 of file torus.cpp.

Function Documentation

◆ _timer()

void _timer ( int value )

Timer callback function to be triggered in at least delay milliseconds.

The number of milliseconds is a lower bound on the time before the callback is generated. GLUT attempts to deliver the timer callback as soon as possible after the expiration of the callback's time interval.

There is no support for canceling a registered callback. Instead, ignore a callback based on its value parameter when it is triggered.

Parameters

value a parameter passed to this callback function.

See also: https://www.opengl.org/resources/libraries/glut/spec3/node64.html

Definition at line 1219 of file torus.cpp.

References _timer(), dateNow(), and delay.

Referenced by _timer(), and main().

◆ cone()

void cone	(	double	h,
		double	r,
		int	vs,
		int	rs,
		int	normals
	)

Draw a cone, centered at the origin, with a circular base on plane XY.

Implicit Equation:

\({x^{2} \over c^{2}}+{y^{2} \over c^{2}} = {z^{2}},\ c = \frac{r}{h}\)
\({f(x,y,z) = \frac{h}{r} x^{2}}+{\frac{h}{r} y^{2}}-{\frac{r}{h} z^{2} = 0}\)
\(\nabla f(x,y,z) = (\frac{h}{r} x,\ \frac{h}{r} y,\ -{\frac{r}{h} z})\)

Parametric Equation:

x(u,v) = u cos(v),
y(u,v) = u sin(v),
z(u,v) = u * h / r,
u ∈ [0,h], v ∈ [0,2Π]
df/du = (cos(v), sin(v), h/r),
df/dv = (-u sin(v), u cos(v) , 0)
df/du x df/dv = (h/r * u cos(v), h/r * u sin(v), -u) = (h/r * x(u,v), h/r * y(u,v), -r/h * z(u,v))

Inverse parametrization:

u = (r/h) * z, or u = (r/h) * (h-z), if cone is upside down.
v = atan2(y,x), if v < 0 then v += 2Π

Note: atan2 returns values ∈ [-Π, Π]

Parameters

h	cone height.
r	cone base radius.
vs	number of divisions in z direction.
rs	number of angular divisions.
normals	whether to draw normals or polygons.

See also: http://mathinsight.org/parametrized_surface_examples; http://mathworld.wolfram.com/Cone.html; https://en.wikipedia.org/wiki/Atan2

Definition at line 242 of file torus.cpp.

References _cone, clamp, getBoxSize(), LEN, scale, TWOPI, and updateBox().

Referenced by init().

◆ cylinder()

void cylinder	(	double	h,
		double	r,
		int	vs,
		int	rs,
		int	normals
	)

Draw a cylinder, centered at the origin, with a circular base on plane XY.

Implicit Equation:

\({x^{2}+y^{2}} = {r^{2}},\) any z
\({f(x,y,z) = x^{2}+y^{2}}-{r^{2} = 0}\)
\({\nabla f(x,y,z) = (x, y, 0)}\)

Parametric Equation:

x(u,v) = r cos(v),
y(u,v) = r sin(v),
z(u,v) = u,
u ∈ [0,h], v ∈ [0,2Π]
df/du = (0, 0, 1),
df/dv = (-r sin(v), r cos(v) , 0)
df/dv x df/du = (r cos(v), r sin(v), 0) = (x(u,v), y(u,v), 0)

Inverse parametrization:

u = z
v = acos(x/r) = atan2(y,x), if v < 0 then v += 2Π

Parameters

h	cylinder height.
r	cylinder radius.
vs	number of divisions in z direction.
rs	number of angular divisions.
normals	whether to draw normals or polygons.

See also: http://mathworld.wolfram.com/Cylinder.html; https://en.wikipedia.org/wiki/Quadric; http://mathworld.wolfram.com/QuadraticSurface.html

Definition at line 313 of file torus.cpp.

References _cylinder, clamp, getBoxSize(), LEN, scale, TWOPI, and updateBox().

Referenced by init().

◆ dateNow()

int dateNow ( )

Return the current Coordinated Universal Time (UTC) in miliseconds.

Returns: time since the Epoch (00:00:00 UTC, January 1, 1970) measured in miliseconds.

Definition at line 1201 of file torus.cpp.

Referenced by _timer().

◆ display()

void display ( void )

Clear window, reset depth buffer and draw torus.

Single buffer does not work, in general.

Definition at line 1141 of file torus.cpp.

References applyTexture, arcball, axis, drawBox(), drawNormals, drawWire, nObjects, objBoxes, OBJECT, objects, rotAngle, showAxes, showBox, spin, theAxes, useArcball, xAxis, yAxis, zAxis, and zobs.

Referenced by main().

◆ drawAxes()

void drawAxes	(	double	r1,
		double	r2
	)

Draw the three coordinate axes.

The length of the axes is r1 + r2.

Parameters

r1	axis length.
r2	axis additional length.

Definition at line 785 of file torus.cpp.

Referenced by init().

◆ drawBox()

void drawBox ( const _BBOX BBOX )

Draw the bounding box as a dashed hexahedron.

Parameters

BBOX	given bounding box.

Definition at line 805 of file torus.cpp.

Referenced by display().

◆ getBoxSize()

double getBoxSize	(	oType	index,
		double *	dx,
		double *	dy,
		double *	dz
	)

Return bounding box dimension.

Parameters

index	Position into objBoxes array.
dx	Bounding box length.
dy	Bounding box width.
dz	Bounding box height.

Returns: Bounding box diameter.

Definition at line 199 of file torus.cpp.

References LEN, and objBoxes.

Referenced by cone(), cylinder(), hyperbolic_paraboloid(), hyperboloid(), paraboloid(), reshape(), sphere(), and torus().

◆ hyperbolic_paraboloid()

void hyperbolic_paraboloid	(	double	w,
		double	h,
		int	ws,
		int	hs,
		int	normals
	)

Draw a hyperbolic paraboloid.

Implicit Equation:

\({xy = z}\)
\({f(x,y,z) = xy - z = 0}\)
\({\nabla f(x,y,z) = (y, x, -1)}\)

Parametric Equation:

x(u,v) = u,
y(u,v) = v,
z(u,v) = uv
u ∈ [0,w], v ∈ [0,h]
df/du = (1, 0, v),
df/dv = (0, 1, u)
df/du x df/dv = (y, x, -1)

Inverse parametrization:

u = x
v = y

Parameters

w	hyperbolic paraboloid width.
h	hyperbolic paraboloid length.
ws	number of divisions in x direction.
hs	number of divisions in y direction.
normals	whether to draw normals or polygons.

See also: http://mathworld.wolfram.com/HyperbolicParaboloid.html; https://en.wikipedia.org/wiki/Quadric; http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html

Definition at line 539 of file torus.cpp.

References _hyperbolic_paraboloid, getBoxSize(), LEN, scale, and updateBox().

Referenced by init().

◆ hyperboloid()

void hyperboloid	(	double	h,
		double	a,
		double	r,
		int	vs,
		int	rs,
		int	normals
	)

Draw a one-sheeted hyperboloid of revolution, centered at the origin, with a circular base parallel to plane XY.

Implicit Equation:

\({{x^{2} \over a^{2}} + {y^{2} \over a^{2}} - {z^{2} \over c^{2}}} = 1\)
\(r = a * \sqrt{1 + \frac{h^2}{4c^2}}\)
\(c = \sqrt{h^2 / (4 (\frac{r^2}{a^2} - 1))},\ r \ge a\)
\({f(x,y,z) = {c^2} x^{2} + {c^2} y^{2} - {a^2} z^{2} - {a^2} {c^2} = 0}\)
\({\nabla f(x,y,z) = (2 \frac{x}{a^2},\ 2 \frac{y}{a^2},\ -2 \frac{z}{c^2})}\)

Parametric Equation:

x(u,v) = a sqrt(1+u²) cos(v),
y(u,v) = a sqrt(1+u²) sin(v),
z(u,v) = c u
u ∈ [-h,h], v ∈ [0,2Π]
df/du = (-a u/srqt(1+u²) cos(v), -a u/srqt(1+u²) sin(v), c),
df/dv = (-a sqrt(1+u²) sin(v), a sqrt(1+u²) cos(v) , 0)
df/dv x df/du = (c x, c y, -a²/c z)

Inverse parametrization:

u = sqrt( x² / a² + y² / a² - 1 )
v = atan2(y,x), if v < 0 then v += 2Π

Parameters

h	hyperboloid height.
a	hyperboloid skirt radius.
r	hyperboloid base radius.
vs	number of divisions in z direction.
rs	number of angular divisions.
normals	whether to draw normals or polygons.

See also: http://mathworld.wolfram.com/One-SheetedHyperboloid.html; http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html; https://rechneronline.de/pi/hyperboloid-e.php

Definition at line 463 of file torus.cpp.

References _hyperboloid, clamp, getBoxSize(), LEN, scale, TWOPI, and updateBox().

Referenced by init().

◆ init()

void init ( void )

Initialize OpenGL state.

Create fifteen display lists for the seven objects, and their normal vectors, plus another one for the coordinate axes.

Use Gouraud shading, and a single light source at the camera position.

OpenGL uses a right-handed coordinate system until the perspective projection, where OpenGL switches to a left-handed coordinate space for the depth buffer (z-buffer) test.

The camera at the origin is looking along -Z axis in eye space,
but it is looking along +Z axis in NDC.
glDepthRange is by default [0, 1] (near, far).
Making the +Z axis point into the screen,
and with +X to the right and +Y up.
It is a left-handed system.
Changing the depth range to [1, 0] will make the system right-handed.

See also: https://www.youtube.com/watch?v=Ck1SH7oYRFM; https://webserver2.tecgraf.puc-rio.br/ftp_pub/lfm/OpenGL_Transformation.pdf; https://www.ics.uci.edu/~gopi/CS211B/opengl_programming_guide_8th_edition.pdf; http://www.songho.ca/opengl/gl_projectionmatrix.html; http://devernay.free.fr/cours/opengl/materials.html; http://math.hws.edu/graphicsbook/c4/s2.html

Definition at line 975 of file torus.cpp.

References cone(), cylinder(), drawAxes(), hyperbolic_paraboloid(), hyperboloid(), initTexture(), nR1, nR2, objects, paraboloid(), R1, R2, R3, sphere(), theAxes, torus(), and zobs.

Referenced by main().

◆ initTexture()

void initTexture ( void )

Initialize texture stuff.

Create a single texture called texName.

See also: https://gregs-blog.com/2008/01/17/opengl-texture-filter-parameters-explained/; https://open.gl/textures

Definition at line 912 of file torus.cpp.

References checkImage, checkImageHeight, checkImageWidth, makeCheckImage(), procImage, and texName.

Referenced by init(), and keyboard().

◆ keyboard()

void keyboard	(	unsigned char	key,
		int	x,
		int	y
	)

Keyboard callback interface.

glutPostRedisplay() marks the current window as needing to be redisplayed. The next iteration through glutMainLoop, the window's display callback will be called to redisplay the window's normal plane.

Parameters

key	key pressed.
x	coordinate of the mouse cursor.
y	coordinate of the mouse cursor.

See also: https://www.opengl.org/resources/libraries/glut/spec3/node49.html

Definition at line 1289 of file torus.cpp.

References applyTexture, arcball, axis, backG, drawNormals, drawWire, ilImage, initTexture(), nObjects, OBJECT, objNames, procImage, reshape(), rotAngle, showAxes, showBox, spin, texName, useArcball, xAxis, yAxis, zAxis, and zobs.

Referenced by main().

◆ loadImage()

int loadImage ( const char * filename )

Load an image from a file.

Parameters

filename file name.

Returns: -1 on fail, or image name on success.

See also: http://openil.sourceforge.net/tuts/tut_4/index.htm

Definition at line 887 of file torus.cpp.

Referenced by main().

◆ main()

int main	(	int	argc,
		char **	argv
	)

Main program for testing.

Arguments:

argc - number of arguments.
argv[0] - Program name.
argv[1] - r1: Distance from the center of the tube to the center of the torus.
argv[2] - r2: Radius of the tube.
argv[3] - r3: Radius in z direction for an ellipsoid.
argv[4] - nr1: number of samples (divisions) on circle r1.
argv[5] - nr2: number of samples (divisions) on circle r2.
argv[6] - fname: texture file name.

Definition at line 1487 of file torus.cpp.

References _timer(), arcball, delay, display(), ilImage, init(), keyboard(), loadImage(), mouseDrag(), mouseFunc(), nR1, nR2, R1, R2, R3, and reshape().

◆ makeCheckImage()

void makeCheckImage ( void )

Create checkerboard texture.

If i's 4th bit is set, i.e., every 8 iterations of i, i & 0x8 yields nonzero.
The same for j. If either is nonzero, but the other is zero, then XOR will yield 1.

This is multiplied by 255, i.e., the maximum value of a channel.

 
   -  i    -> i & 1000
   - 0-7   -> 0
   - 8-15  -> 8
   - 16-23 -> 0
   - 24-31 -> 8
   - ....

See also: https://www.tutorialspoint.com/cprogramming/c_bitwise_operators.htm; https://www.miniwebtool.com/bitwise-calculator/

Definition at line 860 of file torus.cpp.

References checkImage, checkImageHeight, and checkImageWidth.

Referenced by initTexture().

◆ mouseDrag()

void mouseDrag	(	int	_x,
		int	_y
	)

Mouse callback interface.

Set the motion callback for the current window. The motion callback for a window is called when the mouse moves within the window while one or more mouse buttons are pressed.

glutPostRedisplay() marks the current window as needing to be redisplayed. The next iteration through glutMainLoop, the window's display callback will be called to redisplay the window's normal plane.

Parameters

_x	coordinate of the mouse cursor.
_y	coordinate of the mouse cursor.

See also: https://www.opengl.org/resources/libraries/glut/spec3/node51.html

Definition at line 1468 of file torus.cpp.

References arcball, and useArcball.

Referenced by main().

◆ mouseFunc()

void mouseFunc	(	int	state,
		int	button,
		int	_x,
		int	_y
	)

Mouse callback interface.

glutPostRedisplay() marks the current window as needing to be redisplayed. The next iteration through glutMainLoop, the window's display callback will be called to redisplay the window's normal plane.

Parameters

state	mouse button status.
button	mouse button pressed.
_x	coordinate of the mouse cursor.
_y	coordinate of the mouse cursor.

See also: https://www.opengl.org/resources/libraries/glut/spec3/node50.html

Definition at line 1441 of file torus.cpp.

References arcball, and useArcball.

Referenced by main().

◆ paraboloid()

void paraboloid	(	double	h,
		double	r,
		int	vs,
		int	rs,
		int	normals
	)

Draw a paraboloid, centered at the origin, with a circular base on plane XY.

Implicit Equation:

\({x^{2}+y^{2}} = {c\ z},\ c = \frac{r^2}{h}\)
\({f(x,y,z) = \frac{h}{r^2} x^{2} + \frac{h}{r^2} y^{2} - z = 0}\)
\({\nabla f(x,y,z) = (2 \frac{h}{r^2} x,\ 2 \frac{h}{r^2} y,\ - 1)}\)

Parametric Equation:

x(u,v) = u cos(v),
y(u,v) = u sin(v),
z(u,v) = h * (u/r)², (z = h -> u = r)
u ∈ [0,h], v ∈ [0,2Π]
df/du = (cos(v), sin(v), 2*u h/r²),
df/dv = (-u sin(v), u cos(v) , 0)
df/dv x df/du = (2*u h/r² u cos(v), 2*u h/r² u sin(v), -u) = (2h/r² x(u,v), 2h/r² y(u,v), -1)

Inverse parametrization:

u = sqrt(z * r² / h ) or u = sqrt( (h-z) * r² / h ), if paraboloid is upside down.
v = atan2(y,x), if v < 0 then v += 2Π

Parameters

h	paraboloid height.
r	paraboloid radius.
vs	number of divisions in z direction.
rs	number of angular divisions.
normals	whether to draw normals or polygons.

See also: http://mathworld.wolfram.com/Paraboloid.html; https://en.wikipedia.org/wiki/Quadric; http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html

Definition at line 387 of file torus.cpp.

References _paraboloid, clamp, getBoxSize(), LEN, scale, TWOPI, and updateBox().

Referenced by init().

◆ reshape()

void reshape	(	int	w,
		int	h
	)

Handle window resize.

Sets the viewport to its new size.

Parameters

w	viewport width.
h	viewport height.

Definition at line 1254 of file torus.cpp.

References arcball, farPlane, frontPlane, getBoxSize(), OBJECT, toDeg, toRad, viewingAngle, and zobs.

Referenced by keyboard(), and main().

◆ sphere()

void sphere	(	double	r1,
		double	r2,
		double	r3,
		int	numc,
		int	numt,
		int	normals
	)

Draw an ellipsoid or a sphere centered at the origin.

Implicit Equation:

\({x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1,\ a= r_1,\ b = r_2,\ c = r_3\)
\({f(x,y,z) = r_2^2 r_3^2\ x^{2}+r_1^2 r_3^2\ y^{2}+r_1^2 r_2^2\ z^{2}} - (r_1 r_2 r_3)^2 = 0\)
\({\nabla f(x,y,z) = (r_2^2 r_3^2\ x,\ r_1^2 r_3^2\ y,\ r_1^2 r_2^2\ z)}\)

A sphere is defined as the set of all points in three-dimensional Euclidean space R³ that are located at a distance r (the "radius") from a given point (the "center").

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

Parametric Equation:

x(u,v) = r1 cos(u) sin(v),
y(u,v) = r2 sin(u) sin(v),
z(u,v) = r3 cos(v),
u ∈ [0,2Π], v ∈ [0,Π]
df/du = (-r1 sin(u) sin(v), r2 cos(u) sin(v), 0),
df/dv = (r1 cos(u) cos(v), r2 sin(u) cos (v), -r3 sin(v))
df/dv x df/du = (r2 r3/r1 x(u,v), r1 r3/r2 y(u,v), r1 r2/r3 z(u,v)) sin v = (r2² r3² x(u,v), r1² r3² y(u,v), r1² r2² z(u,v))

Inverse parametrization:

u = atan2((y,x) * (r1/r2)), if u < 0 then u += 2Π
v = acos(z/r3)

Parameters

r1	x radius of the ellipsoid.
r2	y radius of the ellipsoid.
r3	z radius of the ellipsoid.
numc	number of divisions along the zero degree longitude (number of latitude divisions).
numt	number of divisions along the zero degree latitude - equator (number of longitude divisions).
normals	whether to draw normals or polygons.

See also: http://mathworld.wolfram.com/Ellipsoid.html; http://mathworld.wolfram.com/Sphere.html; https://en.wikipedia.org/wiki/Ellipsoid; https://sciencing.com/equators-latitude-6314100.html

Definition at line 722 of file torus.cpp.

References _ellipsoid, clamp, getBoxSize(), LEN, scale, TWOPI, and updateBox().

Referenced by init().

◆ torus()

void torus	(	double	r1,
		double	r2,
		int	numc,
		int	numt,
		int	normals
	)

Draw a torus, centered at the origin, and azimuthally symmetric about the z-axis.

In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

Volume: 2 × π² × R × r²
Surface area: 4 × π² × R × r
Common objects with this shape: Doughnut, Ring, Lifebuoy

Implicit Equation:

\((r_1-\sqrt{(x^2+y^2)})^2+z^2=r_2^2,\)
\(f(x,y,z) = (x^2+y^2+z^2+r_1^2-r_2^2)^2 - 4 r_1^2 (x^2+y^2) = 0\), a quartic equation

Parametric Equation:

x(u,v) = (r1 + r2 cos(v)) cos(u),
y(u,v) = (r1 + r2 cos(v)) sin(u),
z(u,v) = r2 sin(v),
u ∈ [0,2Π], v ∈ [0,2Π]
df/du = (-y(u,v), x(u,v), 0),
df/dv = (-z(u,v) cos(u), -z(u,v) sin(u), r2 cos(v))
df/du x df/dv = (r2 x(u,v) cos(v), r2 y(u,v) cos(v), z(u,v) (y(u,v) sin(u) + x(u,v) (cos(u))) = (r2 cos(v) x(u,v), r2 cos(v) y(u,v), (r1 + r2 cos(v)) z(u,v))
df/du x df/dv = (cos(v) cos(u), cos(v) sin(u), sin(v))

Inverse parametrization:

u = atan2(y,x), if u < 0 then u += 2Π
v = asin(z/r2)

When:

r1 > r2, the surface will be the familiar ring torus.
r1 = r2 corresponds to the horn torus, which in effect is a torus with no "hole".
r1 < r2 describes the self-intersecting spindle torus.
r1 = 0, the torus degenerates to the sphere.

Parameters

r1	distance from the center of the tube to the center of the torus.
r2	radius of the tube.
numc	number of divisions of the circle with radius r2.
numt	number of divisions of the circle with radius r1.
normals	whether to draw normals or polygons.

See also: http://mathworld.wolfram.com/Torus.html; https://en.wikipedia.org/wiki/Torus; https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-torus/; https://www.mathsisfun.com/geometry/torus.html; http://trecs.se/torus.php; http://mathworld.wolfram.com/QuarticEquation.html

Definition at line 622 of file torus.cpp.

References _torus, clamp, getBoxSize(), LEN, scale, TWOPI, and updateBox().

Referenced by init().

◆ updateBox()

void updateBox	(	oType	index,
		double	x,
		double	y,
		double	z
	)

Add a new point to a bounding box.

The axis-aligned minimum bounding box (or AABB) for a given point set is its minimum bounding box subject to the constraint that the edges of the box are parallel to the (Cartesian) coordinate axes. It is simply the Cartesian product of N intervals each of which is defined by the minimal and maximal value of the corresponding coordinate for the points in S.

Axis-aligned minimal bounding boxes are used to an approximate location of an object in question and as a very simple descriptor of its shape. For example, in computational geometry and its applications when it is required to find intersections in the set of objects, the initial check is the intersections between their MBBs. Since it is usually a much less expensive operation than the check of the actual intersection (because it only requires comparisons of coordinates), it allows to quickly exclude from checks the pairs that are far apart.

Parameters

index	Position into objBoxes array.
x	Point coordinate.
y	Point coordinate.
z	Point coordinate.

See also: https://en.wikipedia.org/wiki/Minimum_bounding_box

Definition at line 180 of file torus.cpp.

References objBoxes.

Referenced by cone(), cylinder(), hyperbolic_paraboloid(), hyperboloid(), paraboloid(), sphere(), and torus().

Variable Documentation

◆ applyTexture

int applyTexture = 0

Whether to apply texture.

Definition at line 110 of file torus.cpp.

Referenced by display(), and keyboard().

◆ arcball

Arcball arcball

ArcBall.

Definition at line 158 of file torus.cpp.

Referenced by display(), keyboard(), main(), mouseDrag(), mouseFunc(), and reshape().

◆ axis

rAxes axis = xAxis

Current axis.

Definition at line 137 of file torus.cpp.

Referenced by display(), and keyboard().

◆ backG

int backG = 0

Toggle background color.

Definition at line 116 of file torus.cpp.

Referenced by keyboard().

◆ checkImage

GLubyte checkImage[checkImageHeight][checkImageWidth][4]

Procedural image for a checkerboard.

Definition at line 146 of file torus.cpp.

Referenced by initTexture(), and makeCheckImage().

◆ delay

unsigned int delay = 60

Controls the trigger of the timer function.

Definition at line 143 of file torus.cpp.

Referenced by _timer(), and main().

◆ drawNormals

int drawNormals = 0

Whether display vertex normals.

Definition at line 108 of file torus.cpp.

Referenced by display(), and keyboard().

◆ drawWire

int drawWire = 0

Whether draw wireframe or filled polygons.

Definition at line 106 of file torus.cpp.

Referenced by display(), and keyboard().

◆ ilImage

ILuint ilImage

IL image identifier.

Definition at line 150 of file torus.cpp.

Referenced by keyboard(), and main().

◆ nR1

int nR1 = 25

Number of samples.

Definition at line 102 of file torus.cpp.

Referenced by init(), and main().

◆ nR2

int nR2 = 10

Number of samples.

Definition at line 104 of file torus.cpp.

Referenced by init(), and main().

◆ objBoxes

_BBOX objBoxes[nObjects]

Object boxes.

Definition at line 154 of file torus.cpp.

Referenced by display(), getBoxSize(), and updateBox().

◆ OBJECT

int OBJECT = _torus

Current object.

Definition at line 156 of file torus.cpp.

Referenced by display(), keyboard(), and reshape().

◆ objects

GLuint objects

Object display list base index.

Definition at line 94 of file torus.cpp.

Referenced by display(), and init().

◆ objNames

const char* objNames[] = {"Torus", "Cylinder", "Cone", "Ellipsoid", "Paraboloid", "Hyperbolic Paraboloid", "Hyperboloid"}

Object names.

Definition at line 152 of file torus.cpp.

Referenced by keyboard().

◆ procImage

int procImage = 1

Whether to use a procedural texture.

Definition at line 118 of file torus.cpp.

Referenced by initTexture(), and keyboard().

◆ R1

double R1 = 1.0

Distance from the center of the tube to the center of the torus.

Definition at line 96 of file torus.cpp.

Referenced by init(), and main().

◆ R2

double R2 = 0.3

Radius of the tube.

Definition at line 98 of file torus.cpp.

Referenced by init(), and main().

◆ R3

double R3 = 0.5

Radius in z direction for an ellipsoid.

Definition at line 100 of file torus.cpp.

Referenced by init(), and main().

◆ rotAngle

GLfloat rotAngle = 2.0

Rotation angle increment.

Definition at line 140 of file torus.cpp.

Referenced by display(), and keyboard().

◆ showAxes

int showAxes = 0

Whether to show the coordinate axes.

Definition at line 112 of file torus.cpp.

Referenced by display(), and keyboard().

◆ showBox

int showBox = 0

Whether to show the bounding boxes.

Definition at line 114 of file torus.cpp.

Referenced by display(), and keyboard().

◆ spin

int spin = 1

Automatic spin.

Definition at line 122 of file torus.cpp.

Referenced by display(), and keyboard().

◆ texName

GLuint texName

Texture identifier.

Definition at line 148 of file torus.cpp.

Referenced by initTexture(), and keyboard().

◆ useArcball

int useArcball = 0

Whether to use the arcball paradigm.

Definition at line 120 of file torus.cpp.

Referenced by display(), keyboard(), mouseDrag(), and mouseFunc().

◆ viewingAngle

double viewingAngle = 45

Camera viewing angle, also known as opening angle.

Definition at line 92 of file torus.cpp.

Referenced by reshape().

◆ zobs

double zobs = 10.0

Camera position.

Definition at line 90 of file torus.cpp.

Referenced by display(), init(), keyboard(), and reshape().

Data Structures

Macros

Typedefs

Enumerations

Functions

Variables

Detailed Description

Macro Definition Documentation

◆ checkImageHeight

◆ checkImageWidth

◆ clamp

◆ farPlane

◆ frontPlane

◆ GL_GLEXT_PROTOTYPES

◆ LEN

◆ M_PI_

◆ nObjects

◆ posAng

◆ scale

◆ theAxes

◆ toDeg

◆ toRad

◆ TWOPI

Typedef Documentation

◆ _BBOX

◆ oType

◆ Point

◆ rAxes

Enumeration Type Documentation

◆ _oType

◆ _rAxes

Function Documentation

◆ _timer()

◆ cone()

◆ cylinder()

◆ dateNow()

◆ display()

◆ drawAxes()

◆ drawBox()

◆ getBoxSize()

◆ hyperbolic_paraboloid()

◆ hyperboloid()

◆ init()

◆ initTexture()

◆ keyboard()

◆ loadImage()

◆ main()

◆ makeCheckImage()

◆ mouseDrag()

◆ mouseFunc()

◆ paraboloid()

◆ reshape()

◆ sphere()

◆ torus()

◆ updateBox()

Variable Documentation

◆ applyTexture

◆ arcball

◆ axis

◆ backG

◆ checkImage

◆ delay

◆ drawNormals

◆ drawWire

◆ ilImage

◆ nR1

◆ nR2

◆ objBoxes

◆ OBJECT

◆ objects

◆ objNames

◆ procImage

◆ R1

◆ R2

◆ R3

◆ rotAngle

◆ showAxes

◆ showBox

◆ spin

◆ texName