| Last modified : 14 Jul 1999 |
| # 1. System-menu (elementary math) The menu system is separated in two parts. You can define graphics in the plane or in space. A plane graphic only has two dimensions. All graphics in the plane are obtained when you let one variable vary. Graphics in space have three dimensions. These graphics are traced by varying two of the three dimensions. Defining a system consists in determining if the graphic has to vary in two variables or only one. In this case you should choose between the plane or the space. You also need an idea of the minimum and maximum border between which the variables vary. ## a. In the plane The limits between which the variable is situated are defined together with the definition of the system or on the moment of tracing (graph -> describe). This depends from the chosen system. The axes are orthogonal, axes are perpendicular to each other. When you choose [normalise] (in the dialog-box that appears after system->cartesian or system->polar) the system is orthonormal, axes are perpendicular and each axe has the same unity vector. ### 1. Cartesian The cartesian plane is defined by his limites in abscissa (x-axis) and ordinate (y-axis). The limits can be arbitraraly set. But you always have to base your choice on these criteriums : Cartesian analytic The graphic has an analytic form, which means that one of the two variables is a function of the other : y=f(x). The limits of the abscissa (x) will determine the interval in which the function f will vary. So will the limits of the ordinate only be chosen good, when they correspondewith a significatif interval for the variation of the funnction y=f(x). example : y=sin(x) Remark : The function y=sin(x) only has values in the interval [-1, 1] and its periode is 2*PI. In Cartesian-analitic mode it is also possible to derive or integrate nuimeriacally an expression y=f(x). To do this you have to follow next steps. Let us determine the derivate f'(x) of y=sin(x). We all know this will give us y'=cos(x). Go to system->cartesian and give in the limits. Pay attention to what was said above. Now go to system->cartesian->arrow. Let us take D(x) = A'(x). Go to graph->describe and complete the function : y(x)=sin(x). Press on [A]. The functions you see are y=sin(x) and y=cos(x). y=sin(x) is the function descirbed in graph [A]. y=cos(x) is the function descirbed in graph [D]. When you erase graph [D] (graph->describe->arrow) and you draw [A] again (graph->describe and click on "again"-button), only y=sin(x) will be shown. For the integral you can add a constant value to it that corresponds with ex. F(INF(x)) or 0. For this you can give an complex expression from the form g(x) where x represents the inferior limit INF(x), given in abscissa in the system menu. For instance when we want to integrate f(x)=sin(x), the constant we add to the integral can be F(INF(x))=-cos(x). In the general case where we don't know the formel expression of F(x), we choose a constant 0. So the calculated integral will be the following : Integral = F(SUP(x))-F(INF(x)) Remark : To calculate the mean of f(x) in the chosen interval of x, it is sufficiant to divide the calculated integral by the number SUP(x)-INF(x). Cartesian parametric The graphic has a parametric form : x = X(t) y = Y(t) In this case is t the parameter. The required limits for the abscissa and the ordinate depends on the variations of the functions X(t) and Y(t). The variations of the variable t will only be asked once when the system is defined. example : x=cos(t) y=sin(t) This graphic is a circle with centre in (0, 0) and radius r = 1 and can so be drawn in the domain shown on figure 1.1.2. We use an orthonormal system (press the button normalise) and let t varry like : The variation of t has to be one whole tour around the center. If t would only vary from 0 to pi, you would only have a half circle. This is a special case of the more general parametric presentation. x=a*cos(t)+u y=b*sin(t)+v First let us propose that u and v are 0. The parametris representation we have now represents an ellips with a big axis of length 2*a and the length of the small axis 2*b. When a and b are the same you have that the small and big axis of the ellips are the same. This gives us a circle with radius r = a = b. As you can see is a circle a special case of an ellips. The ellips is still centered in the origin (0, 0). This is why we have the numbers u and v. Using u you can position the ellips on the X-axis. When you change the value of v you can position the ellips on the Y-axis. Pay attention you change the limits of the domain. ### 2. Polar The polar coordinate system is defined in abscissa (x) and in ordinate (y) by the definition of an angle t in respect of the half straight line (0x) 0 and a distance r from the point (x, y) to the origin. (0x) 0 is the half straight line with origin (0, 0) and the x represents the positif part of the x-axis. Polar coordinates are defined through the next system : x = r*cos(t) r = sqrt(x^2+y^2) y = r*sin(t) t = atg(y/x) In this coordinate system is the only variable for the graphic the variable t. The limits of the plain can be arbitraraly set. But you always have to base your choice on these criteriums : Polar analytic The graphic is determined by the function r = R(t). To estimate the limits in abscissa and in ordinate, you have to know preliminary what will be the domain in which R(t) will vary in function of the domain of variation of t. example : R(t)=t This is the spiral of Archimedes. For the next interval of variation and a normalised system of axis, this variation interval of t kan be taken : Polar parametric The graph is determined by next system of parametric equations : rho = r(t) theta = q(t) To estimate the limits in abscissa and in ordinate, again you have to know preliminary the interval in which r(t) and q(t) will vary in function of the variation range of t. example : rho(t) = cos(t) theta(t) = sin(t) This will describe us the infinit symbol. We can take the next variation interval : When we choose to normalise the axes and then take for the variation of t : Remark : Analitic polar coordinates are just the same as parametric polar coordinates with only this exception : in parametric polar coordinates you suppose q(t)=t ### 3. 2D images The system of 2d image coordinates permits you to discribe a surface c=P(x,y). c is a level of the colour system of the computer and x & y are respectively the horizontal and the vertical axes. c varies between 0 and the number of colours that can be displmayed by the computer. When c doesn't fit in the interval it is reduced by the modulo of number that can be displayed. ## b. In space In the coordinate systemes in space it is possible to represent surfaces depending on two variables. We have a direct axes system as shown in figure 1.2.1. ### 1. Affine space The represented surfaces are from the form z=f(x,y). It is enough to define the range of variation of the variables x and y, and the limits of the trace of Z. example : sinus cardinal The sinus cardinal (short : sinc) is a function that is defined as follows : sinc = sin(x) / x with "x" a variable Try to describe this function in the plane. Use the parametric cartesian presentation. Take as limits these figures : abscissa [-20, 19] ordinate [-0.4, 1.2] Notice that we have to devide by 0 for x=0 : sin(0)/0 = 0/0 This gives us infinity. That's why we use as upper limit in abscissa 19 and not 20. So we don't get our peak to infinity. The graph can be defined by : The surface [A] will be obtained by introducing the function : In this case is r=sqrt(x^2+y^2). We also could have used the variable t, with t=atg(x/y). ### 2. cylindric coordinates In this coordinate system, it isn possible to represent surfaces s=F(r,t,z). The coordinates are defined by : t stands for Greek letter "theta" To describe a graphic you choose a variation interval for r, theta and z, knowing that that one of the three variables stays constant (a surface depends only of two variables). Then you enter the limits of the space. example : cylinder For this we define r as a constant value and : The limits of space will be : The cylindric surface will be obtained by introducing this function : Once your cylinder is drawn, you can look at it from different angles, simple by rotating it. When you activate the "yes"-button of "hidden faces" and then press the "&dump"-button you will be able to do a Z-buffer rendering. To render the cylinder on a simple Falcon030 it took about 16' 49" with release 09.97. With the release of January 1998 the Z-buffer rendering is accelerated with a factor 4. The same test with this verion only took 03' 09", so it's even faster than the factor 4. Note of the author : Z-buffer rendering is only useful when the surfaces present intersections. ### 3. Spherical coordinates In this coordinate system it is possible to represent surfaces s=F(r,t,p). The coordinates are defined by : t stands for the Greek letter "theta" p stands for theGreek letter "phi" with : x=r*sin(p)*cos(t) y=r*sin(p)*sin(t) z=r*cos(p) To describe a graphic you choose a variation interval for r, theta and phi, knowing that that one of the three variables stays constant (a surface depends only of two variables). Then you enter the limits of the space. example : sphere For this we define r as a constant value and : The limits of space will be : The cylindric surface will be obtained by introducing this function : ### 4. 3D parametrised There exist two parametrised coordinate systems. In both cases only two variables vary. Those two variables can be x and y in a rectangular system or r and t in a polar system. Rectangular In this case we have : r = sqrt(x^2+y^2) t = atg(y/x) The surface that has to be drawn is a function of x, y, r and t. example : torus We define interval of variation like : The limits of space will be : The surface will be obtained by this function : Polar Here we have : x=r*cos(t) y=r*sin(t) The surface to be drawn is a function of x, y, r and t. example : disc For a disc we define the interval of variation : The limits of space will be : Finally we get the disc by introducing this function :