HOW TO SKETCH A CURVE

    Curve sketching has been part of the mathematics syllabus for at
least fifty years. Some people find this enjoyable, and they will try
to sketch curves that the teachers do not necessarily talk about. Every
student of mathematics will at some stage learn about a circle, and how
to draw one. Any one who looks out of the window on a flight over arid
regions will see the occassional circular patch of green. The radius of
the circle is the length of the irrigation pipe which will be moved
around, often by poorly paid migrant workers. This type of irrigation
is called a 'Center Pivot' system.

    <circle.gif +>
    The airline passenger will see all these perfect round shapes and
may be very impressed, but some other curves are involved. The
underground reservoir may have formed over thousands of years, and
modern agricultural methods may deplete this in a short number of
decades. It is possible to make a rough sketch of the amount of water
in that reservoir, against time. The estimation of stuff in underground
reservoirs is very tricky to calculate, and the results are often kept
secret, so quite often the very best graph is nothing but a rough
sketch. There are some important features of such a curve.

    <cliff.gif>
    If the reservoir is replenished by rainfall, faster than it is used
up, then the capacity will increase in time giving a positive slope.
When replenishment and depletion are balanced there will be a zero
slope, and when depletion exceeds replenishment the slope is negative.
The negative slope may appear to be an almost vertical cliff. These
types of cliffs are called discontinuities. In mathematics they are
often expressed as conditional statements such as f(x)= ax if x less
than t and f(x) = at-bx if x greater than t. The C-programming language
also has a construction: <pre> y = (x < t) ? a*x : a*t-b*x; </pre> A
circle can also be expressed as an equation: x^2+y^2=R^2. Here R is the
radius of the circle, and x and y are the cartesian co-ordinates of
points in the plane.

   <quad-0.gif + >
   Other curves are found in text books. The parabola is quite common.
The general equation is y=ax^2+bx+c and this equation may cross the
x-axis at two points, just touch it, or turn away before crossing the
axis. The differential coefficient dy/dx = 2ax+b will always be zero
somewhere (at -b/2a) so there is always a maxinum or minimum.
An example is y=4x(1-x). Here y is zero at x=0 and x=1, and a maximum
at x=1/2. Values of x for which a function f(x)=0 are called roots
of a function, or roots of the equation f(x)=0. The trigonometrical
functions sin(x) and cos(x) have regularly spaced roots at values n*pi
and (n+1/2)*pi. It is easy to see that these functions are essentially
the same with just a shift along the x-axis needed to transform one
into the other.

     <sin-1.gif cos-1.gif>
     The sine function has maxima and minima where cosine is zero.
     <tan-1.gif>
     The tangent function has special problems. The definition tan(x)=
sin(x)/cos(x) means that the function will become infinite whenever cos(x)
=0, and this will happen quite often. The way do do this on the computer
is the use of an <i>implicit</i> function. Find the (x,y) points which
are zeros of a function z=F(x,y) where F is a function of two variables.
Counting zeros of F(x,y) in a given zone does not require computing
values where the function becomes infinity. When the points are sparse
the computer graph may become invisible, but one of the advantages
is that square roots are always given both positive and negative values.

    Two examples are given: a hyperbola y^2-x^2=1 and an elliptic curve
y^2=x(x^2-1). The equation y^2-x^2=1 gives y^2=1+x^2, and there are
no zeros for real values of x. The curve has two distinct parts.
    <hyp-1.gif ec-1.gif>
    These curves are always easier to do with a pencil and paper,
rather than a computer. Mathematica retails for several hundred dollars
and it can do symbolic calculus operations to find tangents, and
radii of curvature. A biro and a few sheets of paper are affordable
in much of the world, except Africa or Afghanistan. A kitchen table
computer system such as the one which produced these sketches is
available for the cost of a google search and download.

       <cubic-1.gif cubic-2.gif>
       Two types of cubic may be seen. One has two distinct turning points
while the other has only a flattening at one point. How a particular curve
y=f(x) behaves can be deduced by examining the <i>differential
coefficient</i> dy/dx = f'(x). For a cubic of the form y=x^3+ax+b this
is f'(x)=3x^2+a with roots + or - sqrt(|a|/3) when a is negative.
<!--STOP
to reconstruct from the text.
PLOT_FN"-o cliff.gif -size 300x200 -lut F. -range 0,10 Y<-(0.3*X*X<9)+(2.7-2.7*X-9)*X>=9"
PLOT_FXY"-o circle.gif -lut F. _3+(X*X)+Y*Y"
PLOT_FN"-o quad-0.gif -size 300x200 -lut F. -range -0.1,1.1 Y<-4*X*1-X"
PLOT_FN"-o sin-1.gif -size 300x200 -lut F. -range -7,7 Y<-1 o X"
PLOT_FN"-o cos-1.gif -size 300x200 -lut F. -range -7,7 Y<-2 o X"
PLOT_FXY"-o tan-1.gif -size 300x200 -lut F. -zone -5,-3,5,3 (1oX)-Y*2oX"
PLOT_FXY"-o hyp-1.gif -size 300x200 -lut F. -zone -3,-2,3,2 _1+(Y*Y)-X*X"
PLOT_FXY"-o ec-1.gif -size 300x200 -lut F. -zone -3,-2,3,2 (Y*Y)-X*_1+X*X"
PLOT_FN"-o cubic-1.gif -size 300x200  -lut F. -range -3,3 Y<-X*X*X"
PLOT_FN"-o cubic-2.gif -size 300x200 -lut F. -range -2,2 Y<-X*_1+X*X"
ess2htm -o sketch.htm -title "SKETCHING A CURVE" sketch.txt
http://www.maths.nuigalway.ie/~chew/teach
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