July 6 2012

(elementary math)

The sytem menu 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 its limits in abscissa (x-axis) and ordinate (y-axis). The limits can be arbitrarily set. But you always have to base your choice on these criterions :
• 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 the limits of the ordinate will only be chosen good, when they correspond with a significant interval for the variations of the function y=f(x).

example : y=sin(x)

Remark :
The function y=sin(x) only has values in the interval [-1, 1] and its period is 2*PI.

In Cartesian-analytic mode it is also possible to derive or integrate numerically an expression y=f(x).
To do this you have to follow next steps.

1. Let us determine the derivate f'(x) of y=sin(x). We all know this will give us y'=cos(x).
2. Go to system->cartesian and give in the limits. Pay attention to what was said above.
3. Now go to system->cartesian->arrow. Let us take D(x) = A'(x).
4. Go to graph->describe and complete the function : y(x)=sin(x).
5. Press on [A].

The functions you see are y=sin(x) and y=cos(x).
y=sin(x) is the function described in graph [A].
y=cos(x) is the function described 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 a 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 formal 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 sufficient 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 t is 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 vary like :

The variations 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 parametrised representation we have now represents an ellipse with a big axis of length 2*a and the length of the small axis 2*b. When a and b are equal, the small and big axis of the ellipse are the same. This gives us a circle with radius r = a = b. As you can see the circle is a special case of an ellipse. The ellipse is still centered in the origin (0, 0). This is why we have the constants u and v. Using u you can position the ellipse on the X-axis. When you change the value of v you can position the ellipse on the Y-axis. Pay attention you change the limits of the domain.

### 2. Polar

The polar coordinates 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 positive part of the x-axis.

Polar coordinates are defined through the following system :

x = r*cos(t)   r = sqrt(x^2+y^2)
y = r*sin(t)    t = atg(y/x)

In this coordinate system, the only variable for the graphic is the variable t. The limits of the plane can be arbitrarily set. But you always have to base your choice on these criterions :

• 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 the domain in which R(t) will vary in function of the domain of variations of t.

example : R(t)=t

This is the spiral of Archimedes.

For the following interval of variations and a normalised system of axis,

this variations interval of t can be chosen :

• Polar parametric

The graph is determined by the 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 variations range of t.

example :

rho(t) = cos(t)
theta(t) = sin(t)

This will describe us the infinite symbol.

We can take the next variations interval :

When we choose to normalise the axes and then take for the variations of t :

Remark :
Analytic polar coordinates are just the same as parametric polar coordinates except that in parametric polar coordinates you suppose q(t)=t

### 3. 2D images

The system of 2d image coordinates allows you to describe 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 displayed 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 systems in space it is possible to represent surfaces depending on two variables. We have the direct axes system as shown in figure 1.2.1.

### 1. Affine space

The represented surfaces are from the form z=f(x,y). You only have to define the range of variations 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 divide 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. cylindrical coordinates

In this coordinate system, it is possible to represent surfaces s=F(r,t,z). The coordinates are defined as :

t stands for Greek letter "theta"

To describe a graphic you choose a variations interval for r, theta and z, knowing that one of the three variables remains 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 cylindrical surface will be obtained by introducing this function :

### 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 the Greek 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 variations interval for r, theta and phi, knowing that one of the three variables remains 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 cylindrical surface will be obtained by introducing this function :

### 4. 3D parametrised

There is two parametrised coordinate systems. In both case 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 variations 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 variations :

The limits of space will be :

Finally we get the disc by introducing this function :