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© François Le Coat
©Karl Samyn
Last modified :
31 Oct 1998
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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)
![[figure 1.1.1]](../images/dumps/figure1.gif)
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.
![[figure 1.1.2]](../images/dumps/figure2.gif)
We use an orthonormal system (press the button normalise) and let t varry like :
![[figure 1.1.3]](../images/dumps/figure3.gif)
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,
![[figure 1.1.4]](../images/dumps/figure4.gif)
this variation interval of t kan be taken :
![[figure 1.1.5]](../images/dumps/figure5.gif)
- 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 :
![[figure 1.1.6]](../images/dumps/figure2.gif)
When we choose to normalise the axes and then take for the variation of t :
![[figure 1.1.7]](../images/dumps/figure3.gif)
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.
![[figure 1.2.1]](../images/dumps/axes_gen.gif)
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.
![[figure 1.2.2]](../images/dumps/axes_aff.gif)
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 :
![[figure 1.2.3]](../images/dumps/figure6.gif)
The surface [A] will be obtained by introducing the function :
![[figure 1.2.4]](../images/dumps/figure7.gif)
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 :
![[figure 1.2.5]](../images/dumps/axes_cyl.gif)
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 :
![[figure 1.2.6]](../images/dumps/figure9.gif)
The limits of space will be :
![[figure 1.2.7]](../images/dumps/figure8.gif)
The cylindric surface will be obtained by introducing this function :
![[figure 1.2.8]](../images/dumps/figure10.gif)
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 :
![[figure 1.2.9]](../images/dumps/axes_sph.gif)
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 :
![[figure 1.2.10]](../images/dumps/figure11.gif)
The limits of space will be :
![[figure 1.2.11]](../images/dumps/figure8.gif)
The cylindric surface will be obtained by introducing this function :
![[figure 1.2.12]](../images/dumps/figure12.gif)
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 :
![[figure 1.2.13]](../images/dumps/figure13.gif)
The limits of space will be :
![[figure 1.2.14]](../images/dumps/figure14.gif)
The surface will be obtained by this function :
![[figure 1.2.15]](../images/dumps/figure15.gif)
- 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 :
![[figure 1.2.16]](../images/dumps/figure16.gif)
The limits of space will be :
![[figure 1.2.17]](../images/dumps/figure8.gif)
Finally we get the disc by introducing this function :
![[figure 1.2.18]](../images/dumps/figure17.gif)
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