Plane curves

Basic functions for drawing curves in a plane are:

Curve given in explicit form y=f(x) is drawn with Plot.
Curve given in implicit form F(x,y)=0 is drawn with ContourPlot.
Curve given in parametric form x=f(t),y=f(t) is drawn with ParametricPlot.

(* Examples *)
Plot[Sin[x],{x,0,2Pi}]
ContourPlot[x^2+y^2==1,{x,-1,1},{y,-1,1}]
ParametricPlot[{t-Sin[t],1-Cos[t]},{t,0,6Pi}]

?Plot
?ContourPlot
?ParametricPlot

Graphical result is an object, that can be given a name to be used later.
The style of a curve (colour,thicknes,..) can be specified with the option PlotStyle.

SinCurve=Plot[Sin[x],{x,0,2Pi},PlotStyle->Red]
CosCurve=Plot[Cos[x],{x,0,2Pi},PlotStyle->{RGBColor[0,1,0],Thick}]

1.a) Plot parametric curve x(t)=4cos((t^1.5)), y(t)=sin(t).  The curve sholud be blue and given a name Slika1.

1.b) Plot the curve y(x)=x/4 sin(5x). The curve sholud be red and given a name Slika2.

1.c) With the function Show show the two curves (Slika1 and Slika2) in the same picture.

2. Epicycloid is a curve traversed by a point on the perimeter of a smaller circle
   that is rolling without sliding on the perimeter of a larger circle.
   Its parametric equation is
     x(φ)=(R+r)cosφ-rcos(((R+r)/r)φ) , y(φ)=(R+r)sinφ-rsin(((R+r)/r)φ)
   where 
     R=radius of the fixed circle
     r=radius of the rolling circle
     φ=polar angle
     
   Draw the epicyclid with suitable radii R and r, the ratio of which should be whole number.
   Unsatisfactory scales in x and y directions can be corrected with the option AspectRatio (see "Help").

Curves in space and surfaces

Three-dimensional objects are drawn with functions Plot3D and ParametricPlot3D.

?Plot3D
?ParametricPlot3D

3.  Draw the curve   ( t cos t,t sin t,t)  with function  ParametricPlot3D  in green colour.
   Find the equation of tangent on this curve at the point (((π (Sqrt[2]))/8),((π (Sqrt[2]))/8),(π/4)).
   Draw the tangent in blue colour and show both objects in the same picture.

4. Find arc length of the curve in exercise 3 between points t=0 and t=7π/11.
   Result: 3.59373

5. a)  Plot the cone  z(x,y)= ((x^2)+(y^2))^(1/2 ) with Plot3D.

     b)  Better looking cone is produced with ParametricPlot3D.
            Plot the same cone using as parameters polar coordinates
	    x=r cos ϕ  and y=r sin ϕ.

6. Plot cylindrical surface   x^2+y^2=1,-3<z<3   with ParametricPlot3D.