tutorial-01

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2007/2008 Graphics Tutorial 1Beforewestart,weneedtodiscusscoordinate systems. In3D,theycomeintwoﬂavors: left-handedand (you guessed it) right-handed. Consider the 2D-coordinate system below:Y321X-3 -2 -1 0 1 2 3-1-2-3This is what we are used to: the vector (1,0), indicating the positive X-direction, points to theright, and (0,1) denotes the positive Y-direction, which is upward.Now, if we want to extend this coordinate system to 3D, then these base vectors become (1,0,0)and (0,1,0), respectively. The third base vector, (0,0,1), is orthogonal to the ﬁrst two, and pointstowards the positive Z-direction. But what direction is that? Coming “out of the paper”, orpointing “into the paper”?Both options are possible. If we consider the XY-plane to be the “ground plane”, then it makessense to consider the Z-direction to be the vertical upward direction (i.e., (0,0,1) comes out of thepaper. On the other hand, if we consider the XY-plane to be aligned with the computer screen,then it makes sense to consider the object that are projected on the screen (while viewing a 3Dmodel) to have positive Z-coordinates, i.e., objects that are farther away have higher Z-values. Inother words, (0,0,1) points into the paper.1A coordinate-system is left-handed if you can point with the thumb, index ﬁnger and middleﬁnger of your left hand into the direction of the positive X-, Y-, and Z-axis, respectively, and it isright-handed otherwise.Problem 1 Are the two ...

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Publié par	Sazef
Nombre de lectures	13
Langue	English

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2007/2008 Graphics Tutorial 1

Before we start, we need to discusscoordinate systems. In3D, they come in two ﬂavors:left-handed and (you guessed it)right-handedthe 2D-coordinate system below:. Consider

This is what we are used to:the vector (1,0), indicating the positive X-direction, points to the right, and (0,1) denotes the positive Y-direction, which is upward.

Now, if we want to extend this coordinate system to 3D, then these base vectors become (1,0,0) and (0,1,The third base vector, (00), respectively.,0,1), is orthogonal to the ﬁrst two, and points towards the positive Z-direction.But what direction is that?Coming “out of the paper”, or pointing “into the paper”?

Both options are possible.If we consider the XY-plane to be the “ground plane”, then it makes sense to consider the Z-direction to be the vertical upward direction (i.e., (0,0,1) comes out of the paper. Onthe other hand, if we consider the XY-plane to be aligned with the computer screen, then it makes sense to consider the object that are projected on the screen (while viewing a 3D model) to have positive Z-coordinates, i.e., objects that are farther away have higher Z-values.In other words, (0,0,1) points into the paper.

A coordinate-system is left-handed if you can point with the thumb, index ﬁnger and middle ﬁnger of your left hand into the direction of the positive X-, Y-, and Z-axis, respectively, and it is right-handed otherwise.

Problem “computer ﬁnger and the other?

1Are the two above-mentioned coordinate systems, based on the “ground plane” and the screen”, respectively, left- or right-handed?Is really true that aligning the thumb, index middle ﬁnger with a given coordinate system is only possible with one hand, and not Try it out (and watch your fellow students make funny movements with their hands).

In graphics, we normally work with two diﬀerent coordinate systems.The XYZ-system is called the “world coordinate system”.This is the coordinate system in which the locations of the objects and the light sources are speciﬁed.The other coordinate system is that of the camera, and it is indicated by UVW (where U can be seen as “the X-axis of the camera, V as its Y-axis, and W as its Z-axis).The origin of the camera is the center of projection (the “view-point”, or the “eye”, so to speak), and we denote it withecoordinates are obviously (0. Its,0,0) in the UVW system, but may be diﬀerent in the XYZ system.

We have to decide for both systems whether we want them to be left-handed or right-handed. Last years textbook (by Slater et al.)adopted the convention of a left-handed world coordinate system, and a right handed viewing coordinate system.The book we use this year (by Shirley) chooses both systems to be right-handed, and that is what we will do in the lectures, tutorials, and programming assignments too.

To simplify things, we tion vanished, and we are going to move our

will initially assume that the XYZ and UVW systems coincide–the distinc-can pretend to have only one (XYZ) coordinate system.Later, when we virtual camera around, we will introduce the UVW system.

Problem 2With our choice of a right-handed coordinate system, does the vector(0,0,1)points into or out of the screen if we consider the computer screen to be aligned with the XY-plane?

Next, we consider theviewing window: thisis a rectangle that is orthogonal to the Z-axis of the camera, and it can be considered as a window through which we view the scene (with our eye ate). The window is fully speciﬁed by ﬁve parameters:lef t,bottom,right,top, andnear. Theﬁrst four specify thex−andy-coordinates of the corners of the window.The ﬁfth parameter determines thez-coordinates of the points in the window (which is equal for all points, since the window is orthogonal to Z-axis).The name of the parameter is derived from the concepts “near plane” and “far plane”, which we will discuss later in the course.Usually, the window is centered around the Z-axis, so thatlef t=−right, andbottom=−top(this is not required, but very common, and we will assume it for now).The value ofnearis negative with our choice of coordinate system (i.e., the Z-axis of the camera points “out of the screen”, and we are looking into the negative direction of the Z-axis).Note that this answers the problem above.. .

A viewing window doesn’t have pixels, but a computer screen does.The number of pixels hor-izontally and vertically are denoted withwidthandheight, respectively.Note that there is no direct relation between these two parameters and the parameters of the camera described above. For instance, if we have a screen with a resolution of 640×480, it doesn’t necessarily mean that lef t=−320,right= 320,bottom=−240, andtopYet, we need to be able to determine= 240. a mapping from pixel centers to the viewing window, in order to be able to compute rays from the origin through the pixels.Such a mapping is called a “windowing transform”.Normally, the aspect ratio of the screen will be the same as that of the viewing window, but this isn’t necessary either (having diﬀerent aspect ratios just means that our image will be distorted).Pixels are

denoted with ordered pairs (i, j), denoting the pixel in thei-th column (from the left) and the jNote that we start counting from 0.-th row (from bottom to top).

Problem 3Given a viewing window with parameterslef t,bottom,right,top, andnear, and a screen with parameterswidthandheight, what are the coordinates of the center of the pixel with indices(i, j)?

In the lecture of last Thursday, we have seen that the parametric equation of a line has the form p+tv, wherepis a point,va vector, andtIfa real value.tis restricted to [0,∞], then the equation denotes the ray in the direction ofv, with originp.

Problem 4Given the camera and screen speciﬁcation of the previous problem, what metric equation of the ray with origine(the origin of the camera, which we assume with the origin of the world coordinate system for the time being) through the center of

is the para-to coincide pixel(i, j)?

The implicit equation of a plane in 3D has the formax+by+cz=d, where (a, b, c,) is anormal vectorwe’ll discuss themof the plane (if you don’t know what normal vectors are, don’t worry:a lotin the near future).

Problem 5What is the equation of the XY-plane?

Problem 6Given a plane with implicit equationax+by+cz=dand a line with parametric equationp+tv, what is the intersection of the plane and the line?

The implicit equation of a sphere in 3D with centerc= (cx, cy, cz) and radiusrhas the form 2 2 22 (x−cx) +(y−cy) +(z−cz) =r.

2 2 22 Problem 7Given a sphere with implicit equation(x−cx) +(y−cy) +(z−cz) =rand a line with parametric equationp+tv, what is the intersection of the sphere and the line?

For some people, it helps to ﬁll in concrete values in the abstract formulas in the previous two problems, to “get a feel” of what is going on.If you are one of those people, then:

Problem 8Deﬁne some “easy” lines, planes, and spheres, redo the previous two problems with conrete parameters, and see if the answers agree with your intuition.

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