The functions of the program

1 norm_vect := proc(v)

 In Parameter :

• vector or list

Out Parameter :

• vector or list

Function :

• return the norm of a vector or a list.

2 tan_vect := proc(v::vector)

 In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return a non unit tangent vector of the vector v.
• If two parameter : return a non unit tangent vector of the vector v, in a precise point.

	

3 tan_unit := proc(v::vector)

 In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return a unit tangent vector of the vector v.
• If two parameter: return a unit tangent vector of the vector v in a precise point.

4 tan_norm := proc(v::vector)

 In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return the norm of a non unit tangent vector of the vector v.
• If two parameter : return the norm of a non unit tangent vector of the vector v, in a precise point.

5 nor_vect := proc(v::vector)

 In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return a non unit normal vector of the vector v.
• If two parameter : return a non unit normal vector of the vector v, in a precise point.

6 curbature := proc(v::vector)

 In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return the norm of a non unit normal vector of the vector v, in other words, the curbature of the vector v.
• If two parameter : return the norm of a non unit normal vector in a precise point, in other words, the curbature of the vector v in a precise point.

7 nor_unit := proc(v::vector)

 In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return a unit normal vector of the vector v.
• If two parameter: return a unit normal vector of the vector v in a precise point.

8 bin_vect := proc(v::vector)

 In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return a non unit bi-normal vector of the vector v.
• If two parameter : return a non unit bi-normal vector of the vector v, in a precise point.

9 torsion := proc(v::vector) 

In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return the norm of a non unit bi-normal vector of the vector v, in other words, the torsion of the vector v.
• If two parameter : return the norm of a non unit bi-normal vector in a precise point, in other words, the torsion of the vector v in a precise point.

10 bin_unit := proc(v::vector) 

In Parameter :

• vector
• vector,point

Out Parameter :

• vector

Function :

• If one parameter : return a unit bi-normal vector of the vector v.
• If two parameter: return a unit bi-normal vector of the vector v in a precise point.

11 kap1 := proc(k,twist) 

In Parameter :

• curbature, twist
• curbature, twist, point

Out Parameter :

• kappa1

Function :

• If two parameter : return the value of kappa1 for all points
• If three parameter: return the value of kappa1 for a precise point.

12 kap2 := proc(k,twist) 

In Parameter :

• curbature, twist
• curbature, twist, point

Out Parameter :

• kappa2

Function :

• If two parameters : return the value of kappa2 for all points
• If three parameter: return the value of kappa2 for a precise point.

13 kap3 := proc(t,twist) 

In Parameter :

• curbature, twist
• curbature, twist, point

Out Parameter :

• kappa3

Function :

• If two parameters : return the value of kappa3 for all points
• If three parameters: return the value of kappa3 for a precise point.

14 d1 := proc(v::vector,twist) 

In Parameter :

• position vector, twist
• position vector, twist, point

Out Parameter :

• vector d1

Function :

• If two parameter : return the vector d1 for all points.
• If three parameter: return the vector d1 for a precise point.

15 d2:= proc(v::vector,twist) 

In Parameter :

• position vector, twist
• position vector, twist, point

Out Parameter :

• vector d2

Function :

• If two parameter : return the vector d2 for all points.
• If three parameter: return the vector d2 for a precise point.

16 d3 := proc(v::vector) 

In Parameter :

• position vector
• position vector, point

Out Parameter :

• vector d3

Function :

• If one parameter : return the vector d3 for all points.
• If two parameter: return the vector d3 for a precise point.

17 kappa_to_k := proc(k1,k2) 

In Parameter :

• kappa1, kappa2

Out Parameter :

• curbature

Function :

• return the value of the curbature from the values of kappa1 and kappa2.

18 kappa_to_dn := proc(k1,k2,k3) 

In Parameter :

• kappa1, kappa2, kappa3

Out Parameter :

• table created as follow : table[1] represent the vector d1, table[2] represent the vector d2 and table[3] represent the vector d3.

Function :

• Find the three direction vectors d1, d2 et d3 from k1, k2 et k3. These vectors must be found numerically. So if we do d_vect := kappa_to_dn_di(…), d_vect[1] would represent d1, d_vect[2] d2 and d_vect[3] d3. d_vect[1](1) would calculate d1 in the point where s = 1.

19 torcur_to_tnb_di := proc(k,tor) 

In Parameter :

• Courbature, torsion

Out Parameter :

• table created as follow : table[1] represent the unit tangent vector , table[2] represent the unit normal vector and table[3] represent the unit bi-normal vector.

Function :

• Find the three direction vectors t, n et b from the curbature and the torsion in a point. These vectors are found numerically. So if we do d_vect := torcour_to_tnb_di(…), d_vect[1] would represent 1_t, d_vect[2] 1_n and d_vect[3] 1_b. d_vect[1](1) would calculate 1_t in the point where s = 1.

20 discret_p := proc(v::vector,x,max,h,twist) 

In Parameter :

• position vector, start, maximum, number of points, twist

Out Parameter :

• table created as follow : table[i][1] represent the position vector

  table[i][2] represent the unit tangent vector

  table[i][3] represent the unit normal vector

  table[i][4] represent the unit bi-normal vector

  table[i][5] represent the curbature

  table[i][6] represent the torsion.

  table[i][7] represent the norm of the tangent vector

  table[i][8] represent kappa1

  table[i][9] represent kappa2.

  table[i][10] represent kappa3.

  table[i][11] represent the twist at a precise point

  table[i][12] represent d1.

  table[i][13] represent d2.

  table[i][14] represent d3.

The i represents all the points discretisized from 1 to h. So table[1][1] represent the position vector of the vector v given in parameter at the point x, table[2][1] represent the position vector of the vector v at the point x+(max-x)/h, …

Function :

• Find all the important characteristics of a vector for a given number of points in a certain interval.

21 discret_p_tnb := proc(k1,k2,x,max,h) 

In Parameter :

• curbature, torsion, start, maximum, number of points, twist

Out Parameter :

• table created as follow : table[i][1] represent the position vector

  table[i][11] represent the twist at a precise point

The i represents all the points discretisized from 1 to h. So table[1][1] represent the position vector of the vector v given in parameter at the point x, table[2][1] represent the position vector of the vector v at the point x+(max-x)/h, …

Function :

• Find the values of the position vector and the twist of a filament for a given number of points in a certain interval

22 discret_p_tnb := proc(k1,k2,x,max,h) 

In Parameter :

• curbature, torsion, start, maximum, number of points, twist

Out Parameter :

• table created as follow : table[i][1] represent the unit tangent vector

  table[i][2] represent the unit normal vector

  table[i][3] represent the unit bi-normal vector.

The i represents all the points discretisized from 1 to h. So table[1][1] represent the position vector of the vector v given in parameter at the point x, table[2][1] represent the position vector of the vector v at the point x+(max-x)/h, …

Function :

• Find the values of t, n et b of a vector for a given number of points in a certain interval.

23 discret_p_kappa := proc(k1,k2,k3,x,max,h) 

In Parameter :

• kapp1, kapp2, kappa3, start, maximum, number of points

Out Parameter :

• table created as follow : table[i][1] represent the unit tangent vector

  table[i][2] represent the unit normal vector

  table[i][3] represent the unit bi-normal vector.

The i represents all the points discretisized from 1 to h. So table[1][1] represent the position vector of the vector v given in parameter at the point x, table[2][1] represent the position vector of the vector v at the point x+(max-x)/h, …

 Function :

• Find the values of t, n et b of a vector for a given number of points in a certain interval.

24 dna_filament := proc(v::vector,x,max,radi,lambda) 

In Parameter :

• position vector, start, maximum, width of the tube, twist

Out Parameter :

• none

Function :

• Draw the vector v in the plane, and a filament surrounding him with a given twist and in a certain interval. The vectors used are 1_n and 1_b

25 dna_d_filament := proc(v::vector,x,max,radi,lambda) 

In Parameter :

• position vector, start, maximum, width of the tube, twist

Out Parameter :

• none

Function :

• Draw the vector v in the plane, and a filament surrounding him with a given twist and in a certain interval. The vectors used are d1 and d2

26 funky_filament := proc(v::vector,x,max,radi,lambda) 

In Parameter :

• position vector, start, maximum, width of the tube, twist

Out Parameter :

• none

Function :

• Draw the vector v in the plane, and a filament surrounding him with a width which decrements as a negative exponential, with a given twist and in a certain interval.

27 any_filament := proc(v::vector,x,max,radi,form_1,form_2) 

In Parameter :

• position vector, start, maximum, width of the tube, form of the vector 1_n, form of the vector 1_b.

Out Parameter :

• none

Function :

• Draw the vector v in the plane, and a filament surrounding him which can take any form, and in a certain interval. The two parameters form_1 and form_2 must be cautiously created in order to create any filament. The procedures Square1 and Square2 given later are used to create square filaments.

28 ressort_filament := proc(v::vector,x,max,radi,twist) 

In Parameter :

• position vector, start, maximum, width of the tube, twist.

Out Parameter :

• none

Function :

• Used to create a rectiligne filament and a filament which surrounds it, with a given twist and in a certain interval.

29 dna2_filament := proc(v::vector,x,max,radi,twist) 

In Parameter :

• position vector, start, maximum, width of the tube, twist.

Out Parameter :

• none

Function :

• Used to create a filament, a filament which surrounds it, and another one surrounding this last one, this with a given twist and in a certain interval.

30 tor_filament := proc(v::vector,x,max,radi,twist) 

In Parameter :

• position vector, start, maximum, width of the tube, twist.

Out Parameter :

• none

Function :

• Used to create a filament which resembles a snake.

31 tube_filament := proc(v::vector,x,max,radi,twist) 

In Parameter :

• position vector, start, maximum, width of the tube, twist.

Out Parameter :

• none

Function :

• Used to create a tube and an information on its twist.

32 point_filament := proc(v::vector,x,max,h,twist) 

In Parameter :

• position vector, start, maximum, number of points, twist.

Out Parameter :

• none

Function :

• Used to draw a given filament and a filament surrounding him with a given twist and in a certain interval, these filaments being discretised, and so, being drawn point by point.

33 square1 := proc(smax,twist) 

In Parameter :

• maximum of the interval, twist

Out Parameter :

• In collaboration with Square2, these functions output two functions which works together and defines a square in space with the increment of the parameter s.

Function :

• Used to define the form of the filament to be a square form. This function modify one of the unit vector : 1_b, 1_n, d1 or d2 and the other function will modify the other one in the same family ((1_b,1_n),(d1,d2)).

34 square2 := proc(smax,twist) 

In Parameter :

• maximum of the interval, twist

Out Parameter :

• In collaboration with Square2, these functions output two functions which works together and defines a square in space with the increment of the parameter s.

Function :

• Used to define the form of the filament to be a square form. He will modify the twin unit vector of the ones of Square1.

35 tan_points := proc(l::list) 

In Parameter :

• List of points.

Out Parameter :

• List composed of the values for the unit tangent vectors and the tangent vectors in these points

Function :

• Used to create a list of unit tangent vectors in precise points from a discretisized curve.

36 nor_cur_points := proc(l:list) 

In Parameter :

• List of points.

Out Parameter :

• List composed of the values for the unit normal vectors, the curvatures, the kappa1 and the kappa2 in these points.

Function :

• Used to create a list containing the unit normal vectors, the curvatures, the kappa1 and the kappa2 in precise points of a discretisized curve.

37 bin_tor_points := proc(l:list) 

In Parameter :

• List of points.

Out Parameter :

• List composed of the values for the unit bi-normal vectors, the torsion, the kappa3, and the directions vectors d₁ and d₂, in these points.

Function :

• Used to create a list containing the unit bi-normal vectors, the torsion, the kappa3, the directions vectors d₁ and d₂ of a discretisized curve.

38 put_in := proc(aa,x,max,hh) 

In Parameter :

• Table created with discret_p, start of the interval, maximum, number of points of the interval.

Out Parameter :

• List composed of the points of the curve, of the torsion’s in each point and of the value of s for each points.

Function :

• Used to make the transition between the global table created with discret_p, and the other functions by taking in account only the points necessary for the discretisation of the curve and for his analysis.

39-57 are functions which works internally by numerical means, they do not calculate any equation symbolically; they work only, and directly, from a list of discretized points created out of the position vector in a certain interval.

39 glob_dis := proc(v::vector,xx,maxx,hh,twist)

 In Parameter :

• Vector, start, maximum, number of points,twist.

Out Parameter :

• List composed of the points of the curve, the values for the direction vectors d₁ at these points, the values for the direction vector d₂ at these points.

Function :

• Used to calculate the values for the points of the curve, d₁ and d₂ in one time, from the equation of a vector, the twist function and an interval with the total number of points to be discretisized.

40 glob_dis2 := proc(l) 

In Parameter :

• List composed of points, values of the local twist for each point, value of the arclength parameter for each point.

Out Parameter :

• List composed of the points of the curve, the values for the direction vectors d₁ at these points, the values for the direction vector d₂ at these points.

Function :

• Used to calculate the values for the points of the curve, d₁ and d₂ in one time, from a list of points, twist and arclength values.

41 point_dis := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the vector in each points of the specified interval

Function :

• Used to create a list of the points of the vector in a certain interval of a curve.

42 point_dis2 := proc(l)

 In Parameter :

• List composed of points, values of the local twist for each point, value of the arclength parameter for each point.

Out Parameter :

• List composed of the points of the curve

Function :

• Used to calculate the values for the points of the curve, from a list of points, twist and arclength values.

43 tan_dis := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the unit tangent vectors in each points of the specified interval

Function :

• Used to create a list of unit tangent vectors in a certain interval of a curve.

44 tan_dis2 := proc(v) 

In Parameter :

• List composed of points.

Out Parameter :

• List composed of the unit tangent points of the curve represented by the list of points

Function :

• Used to calculate the values for the unit tangent points of a discretisized curve.

45 nor_dis := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the unit normal vectors in each points of the specified interval

Function :

• Used to create a list of unit normal vectors in a certain interval of a curve.

46 bin_dis := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the unit bi-normal vectors in each points of the specified interval

Function :

• Used to create a list of unit bi-normal vectors in a certain interval of a curve.

47 cur_dis := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the curbature in each points of the specified interval

Function :

• Used to create a list of curbature in a certain interval of a curve.

48 tor_dis := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the torsion in each points of the specified interval

Function :

• Used to create a list of torsion’s in a certain interval of a curve.

49 kap1_dis := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the kappa1 in each points of the specified interval

Function :

• Used to create a list of kappa1 in a certain interval of a curve.

50 kap2_dis := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the kappa2 in each points of the specified interval

Function :

• Used to create a list of kappa2 in a certain interval of a curve.

51 kap3_dis := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the kappa3 in each points of the specified interval

Function :

• Used to create a list of kappa3 in a certain interval of a curve.

52 kap3_dis2 := proc(l) 

In Parameter :

• List composed of points, values of the local twist for each point, value of the arclength parameter for each point.

Out Parameter :

• List composed of the values for Kappa3 of a discretisized curve

Function :

• Used to calculate the values for Kappa3 of a discretisized curve.

53 d1_dis := proc(v::vector,xx,maxx,hh,twist) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the direction vector d1 in each points of the specified interval

Function :

• Used to create a list of the values for the direction vector d₁ in a certain interval of a curve.

54 d1_dis2 := proc(l) 

In Parameter :

• List composed of points, values of the local twist for each point, value of the arclength parameter for each point.

Out Parameter :

• List composed of the values for the direction vector d₁ of a discretisized curve

Function :

• Used to calculate the values for the direction vector d₁ of a discretisized curve.

55 d2_dis := proc(v::vector,xx,maxx,hh,twistt)

 In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the direction vector d₂ in each points of the specified interval

Function :

• Used to create a list of the values for the direction vector d₂ in a certain interval of a curve.

56 d2_dis2 := proc(l) 

In Parameter :

• List composed of points, values of the local twist for each point, value of the arclength parameter for each point.

Out Parameter :

• List composed of the values for the direction vector d₂ of a discretisized curve

Function :

• Used to calculate the values for the direction vector d₂ of a discretisized curve.

57 d3_dis := proc(v::vector,xx,maxx,hh,twistt)

 In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the direction vector d₃ in each points of the specified interval

Function :

• Used to create a list of the values for the direction vector d₃ in a certain interval of a curve.

58-69 are function which works by finding the needed curve’s properties by symbolic means out of the position vector.

58 point_dis3 := proc(v::vector,xx,maxx,hh,twist)

 In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the vector in each points of the specified interval

Function :

• Used to create a list of the points of the vector in a certain interval of a curve.

59 tan_dis3 := proc(v::vector,xx,maxx,hh)

 In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the unit tangent vectors in each points of the specified interval

Function :

• Used to create a list of unit tangent vectors in a certain interval of a curve.

60 nor_dis3 := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the unit normal vectors in each points of the specified interval

Function :

• Used to create a list of unit normal vectors in a certain interval of a curve.

61 bin_dis3 := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the unit bi-normal vectors in each points of the specified interval

Function :

• Used to create a list of unit bi-normal vectors in a certain interval of a curve.

62 tor_dis3 := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the torsion in each points of the specified interval

Function :

• Used to create a list of torsion’s in a certain interval of a curve.

63 cur_dis3 := proc(v::vector,xx,maxx,hh) 

In Parameter :

• Vector, start, maximum, number of points.

Out Parameter :

• List of the values for the curbature in each points of the specified interval

Function :

• Used to create a list of curbature in a certain interval of a curve.

64 kap1_dis3 := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the kappa1 in each points of the specified interval

Function :

• Used to create a list of kappa1 in a certain interval of a curve.

65 kap2_dis3 := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the kappa2 in each points of the specified interval

Function :

• Used to create a list of kappa2 in a certain interval of a curve.

66 kap3_dis3 := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the kappa3 in each points of the specified interval

Function :

• Used to create a list of kappa3 in a certain interval of a curve.

67 d1_dis3 := proc(v::vector,xx,maxx,hh,twist) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the direction vector d1 in each points of the specified interval

Function :

• Used to create a list of the values for the direction vector d₁ in a certain interval of a curve.

68 d2_dis3 := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the direction vector d₂ in each points of the specified interval

Function :

• Used to create a list of the values for the direction vector d₂ in a certain interval of a curve.

69 d3_dis3 := proc(v::vector,xx,maxx,hh,twistt) 

In Parameter :

• Vector, start, maximum, number of points, twist.

Out Parameter :

• List of the values for the direction vector d₃ in each points of the specified interval

Function :

• Used to create a list of the values for the direction vector d₃ in a certain interval of a curve.

70 fragme1 := proc(x,y,total) 

In Parameter :

• List of values, list of values, total number of points

Out Parameter :

• List of points.

Function :

• Used to create a list of points representing the error between the division of x and y. If the division is 1, they are equal, if 0.5, the value of the list x considered for a given index is twice as big as the y’s value for the same index. We can then draw this division for each point and have a global view of the relative error.

 71 fragme2 := proc(x,y,level,total)

 In Parameter :

• List of vectors, list of vectors, axe of the vectors of the list x to be considererd, total number of points.

Out Parameter :

• List of points.

Function :

• Used to create a list of points representing the error between the division of x and y. If the division is 1, they are equal, if 0.5, the value of the list x considered for a given index is twice as big as the y’s value for the same index. We can then draw this division for each point and have a global view of the relative error.

72 error_discret := proc(v::vector,x,max,h,twist) 

In Parameter :

• position vector, start, maximum, number of points, twist.

Out Parameter :

• none

Function :

• Used to have the error curves of the discretisized values compared to the real values.

73 Writhe1 := proc(v,x,max,delta2)

In Parameter :

• position vector, start, maximum, number of points.

Out Parameter :

• real

Function :

• Used to calculate the Writhing number of a knot by using the Gauss Writhing formula. It works by finding the needed curve’s properties in the function by numerical means from the discretizing of the position vector in the given interval. It works faster than the function Writhe2 but is less accurate.

74 Writhe2 := proc(v,x,max,delta2) 

In Parameter :

• position vector, start, maximum, number of points.

Out Parameter :

• real

Function :

• Used to calculate the Writhing number of a knot using the Gauss Writhing formula. It works by finding the needed curve’s properties in the function by symbolic means from the position vector. It is therefore more accurate than Writhe1 but is also slower.

75 Link1 := proc(v1,v2,x,max,delta2)

 In Parameter :

• position vector, position vector, start, maximum, number of points.

Out Parameter :

• real

Function :

• Used to calculate the Linking number of two knots using the Gauss Linking formula. It works by finding the needed curve’s properties in the function by numerical means from the discretizing of the position vector in the given interval. It works faster than the function Link2 but is less accurate.

76 Link2 := proc(v1,v2,x,max,delta2) 

In Parameter :

• position vector, position vector, start, maximum, number of points.

Out Parameter :

• real

Function :

• Used to calculate the Linking number of two knots using the Gauss Linking formula. It works by finding the needed curve’s properties in the function by symbolic means from the position vector. It is therefore more accurate than Link1 but is also slower.

77 Twist1 := proc(v,x,max,delta2,twistfunction)

 In Parameter :

• position vector, start, maximum, number of points, twist function.

Out Parameter :

• real

Function :

• Used to calculate the Twist value of a curve. It works by finding the needed curve’s properties in the function by numerical means from the discretizing of the position vector in the given interval. It works faster than the function Twist2 but is less accurate.

78 Twist2 := proc(v,x,max,delta2,twistfunction)

 In Parameter :

• position vector, start, maximum, number of points, twist function.

Out Parameter :

• real

Function :

• Used to calculate the Twist value of a curve. It works by finding the needed curve’s properties in the function by symbolic means from the position vector. It is therefore more accurate than Twist1 but is also slower.

79 Torsion1 := proc(v,x,max,delta2)

 In Parameter :

• position vector, start, maximum, number of points.

Out Parameter :

• real

Function :

• Used to calculate the total torsion of a curve. It works by finding the needed curve’s properties in the function by numerical means from the discretizing of the position vector in the given interval. It works faster than the function Torsion2 but is less accurate.

80 Torsion2 := proc(v,x,max,delta2)

 In Parameter :

• position vector, start, maximum, number of points.

Out Parameter :

• real

Function :

• Used to calculate the total torsion of a curve. It works by finding the needed curve’s properties in the function by symbolic means from the position vector. It is therefore more accurate than Torsion1 but is also slower.

81-117 are maple to pov conversion procedures.

81 mheader := proc()

 In Parameter :

• none

Out Parameter :

• none

Function :

• Used to create the header.

82 minclude := proc(mlist) 

In Parameter :

• list of strings

Out Parameter :

• none

Function :

• Used to create all the includes needed.

83 mcamera := proc(t)

 In Parameter :

• none
• list of list

Out Parameter :

• none

Function :

• no parameter : default camera
• one parameter : user defined camera. For examples [[0,4,-100],[0,0,0]] creates a camera located at [0,4,-100] which looks at [0,0,0].

84 mlight := proc(mloc::list) 

In Parameter :

• list

Out Parameter :

• none

Function :

• Used to create a white light spot at the parameter place. For example [0,100,0] create a light spots at [0,100,0].

85 mspotlight := proc(mloc::list,mat::list,mrad,mfall,mtight) 

In Parameter :

• list, list, radius, fallout, tightness

Out Parameter :

• none

Function :

• Used to create a spotlight centered at mloc, pointing at mat, with radius mrad, fallout mfall and tightness mtight.

86 mplane := proc(mtype)

 In Parameter :

• integer (1-9)

Out Parameter :

• none

Function :

• Used to create a plane, there are currently 9 types of plane.

87 mbackground := proc(color)

 In Parameter :

• string

Out Parameter :

• none

Function :

• Used to create colored background.

88 msky := proc(mtype)

 In Parameter :

• integer (1-3)

Out Parameter :

• none

Function :

• Used to create a sky, there are currently 3 types of background.

 89 minit := proc(mname)

 In Parameter :

• string

Out Parameter :

• none

Function :

• Used to decide which name the POV output file will take.

90 mfinish := proc() 

In Parameter :

• none

Out Parameter :

• none

Function :

• Used to signal that we are no longer interested in writing the output of maple into the POV file.

91 mdeclare := proc(mdec)

 In Parameter :

• string

Out Parameter :

• none

Function :

• Used to declare a variable.

92 mwhile := proc(mco) 

In Parameter :

• string

Out Parameter :

• none

Function :

• Used to create a while « condition ».

93 mend := proc()

In Parameter :

• none

Out Parameter :

• none

Function :

• Used to terminate a « while » or an « if »

94 mif := proc(mco)

In Parameter :

• string

Out Parameter :

• none

Function :

• Used to create an if « condition ».

95 melse := proc()

 In Parameter :

• none

Out Parameter :

• none

Function :

• Used to create an else.

96 mtexture := proc(mcolor)

 In Parameter :

• integer

Out Parameter :

• none

Function :

• Used to create a texture with a specified color.

97 msphere := proc(mcenter,mradius,mcolor)

 In Parameter :

• list, integer, integer

Out Parameter :

• none

Function :

• Used to create a sphere with center mcenter, radius mradius and color mcolor.

98 mbox := proc(mcorner1,mcorner2,mcolor)

 In Parameter :

• list, list, integer

Out Parameter :

• none

Function :

• Used to create a rectangular box with one corner being mcorner1, the other corner being mcorner2, and the color being mcolor.

99 mcone := proc(mend1,mend2,mrad1,mrad2,mcolor)

 In Parameter :

• list, list, integer, integer, integer

Out Parameter :

• none

Function :

• Used to create a cone with one end being mend1, the other end being mend2, the radius at mend1 being mrad1, the radius at mend2 being mrad2 and the color being mcolor.

100 mcyl := proc(mend1,mend2,mradius,mcolor)

 In Parameter :

• list, list, integer, integer

Out Parameter :

• none

Function :

• Used to create a cylinder with one end being mend1, the other end being mend2, the width of the cylinder being mradius, and its color being mcolor.

101 munion := proc()

 In Parameter :

• none

Out Parameter :

• none

Function :

• Used to create a union, it must end with mend_csg().

102 mintersection := proc()

 In Parameter :

• none

Out Parameter :

• none

Function :

• Used to create an intersection, it must end with mend_csg().

103 mdifference := proc()

 In Parameter :

• none

Out Parameter :

• none

Function :

• Used to create a difference, it must end with mend_csg().

104 mmerge := proc()

 In Parameter :

• none

Out Parameter :

• none

Function :

• Used to create a merge, it must end with mend_csg().

105 mend_csg := proc()

 In Parameter :

• none

Out Parameter :

• none

Function :

• Used to end the above.

106 mtranslate := proc(mtrans)

 In Parameter :

• list

Out Parameter :

• none

Function :

• Used to create a translation.

107 mrotate := proc(mrotate)

 In Parameter :

• list

Out Parameter :

• none

Function :

• Used to create a rotation.

108 mscale := proc(mscale)

 In Parameter :

• list

Out Parameter :

• none

Function :

• Used to create a scaling.

109 mambient_light := proc(mcolor)

 In Parameter :

• integer

Out Parameter :

• none

Function :

• Used to create an ambient_light of color mcolor.

110 mtext := proc(mte,mcolor,mthick,moff)

 In Parameter :

• string, integer, integer, integer

Out Parameter :

• none

Function :

• Used to create text. The text is mte, mcolor is the color of the text, mthick is thickness of the characters, and moff is the spacing between the characters.

111 mscene := proc(mnum)

 In Parameter :

• integer (1-17)

Out Parameter :

• none

Function :

• Used to create a scene, there are currently 17 different scenes to choose from.

112 manim := proc(mframe,mname)

 In Parameter :

• integer, string

Out Parameter :

• none

Function :

• Used to create an animation file for the POV file mname, and with mframe number of frames.

113 mknot := proc(mnum1,mnum2)

 In Parameter :

• integer, integer

Out Parameter :

• list

Function :

• This procedure takes two parameters, the first giving the number of crossings of the knot, the second, its rank among all knots having this number of crossings, and gives out a list of points representing the harmonic knots. For example mknot(3,1) returns: [41,-18,-83,-83,-11,27,36,27,-113,30,11,-27,0,45,-30,113,-11,27], which means the harmonic knot equation is : [41cos(s)-18sin(s)-83cos(2s)-83sin(2s)-11cos(3s)+27 sin(3s),36cos(s)+27sin(s)-113cos(2s)+30sin(2s)+11cos(3s)-27sin(3s), 0cos(s)+ 45sin(s) -30cos(2s)+113sin(2s)-11cos(3s)+27sin(3s)]. This is the trefoil knot.

114 knot := proc(mnum1,mnum2)

 In Parameter :

• integer, integer

Out Parameter :

• vector

Function :

• This procedure takes two parameters, the first giving the number of crossings of the knot, the second, its rank among all knots having this number of crossings, and gives out the knot’s vector equation. For example mtransf(3,1) returns [41cos(s)-18sin(s)-83cos(2s)-83sin(2s)-11cos(3s)+27 sin(3s),36cos(s)+27sin(s)-113cos(2s)+30sin(2s)+11 cos(3s)-27sin(3s), 0cos(s)+ 45sin(s) -30cos(2s)+113sin(2s)-11cos(3s)+27sin(3s)]. This is the trefoil knot.

115 mshowscenes := proc(mname)

 In Parameter :

• string

Out Parameter :

• 17 POV files

Function :

• This function receive as parameter a name. It will create 17 files, of the form mname01, mname02, … , mname17 with .pov extension. These files represents the 17 different scenes available in this package, all of them have a 7/13 torus knot image on them. This enables one to actually see all of the available scenes.

116 surfaceplot := proc(param1,param2,param3)

 In Parameter :

• equation, list of list, list

Out Parameter :

• list of list

Function :

• This function takes 3 parameters. The first one represents the equation of any surface, for example : z^2+y^2+x^2 = 1, the second parameters gives the interval for the x and y variables in the following form: [[x_left,x_right],[y_left,y_right]], the third interval is a list of two numbers specifiying respectively the number of points for the x interval and for the y interval by which this equation is going to be discretized. The result of this function is a list of points representing this surface. This can then be transfered to POV using the main, povdisplay, function.

117 povdisplay := proc(param1,param2,param3)

 In Parameter :

• string, equation/list/list of list/vector, list/integer

Out Parameter :

• 1 or 2 POV file(s)

Function :

• This function is a Maple to POV converter. 

List of parameters(mandatory):

	 First parameter(type)

	

surface - draws a surface Z(X,Y)

knot - draws a knot

curve - draws a curve(not discretisised)

vector - draws a curve(discretisised)

points - draws a list of points

Second parameter(equations)

surface - equation of a surface in the form Z(X,Y)

knot - [number of crossings, rank in its familly]

curve - equation of a curve with parameter s (example: vector([cos(s),s,sin(s)]))

vector - equation of a curve with parameter s

points - list of points in the form of a list of list

points+twist - list of the points coordinates + local twist at each point + value of

			 the parameter s for each point

Third parameter(interval)

surface - interval for the X and Y parameters(example: [[0,6],[0,6]])

knot - interval for the s parameter(example: [0,6])

curve - interval for the s parameter

vector - interval for the s parameter

points - value must be 1

points+twist - value must be 2

List of options(all have default values):

orientation - gives the orientation of the resulting picture(example: orientation=[

	 -90,90], default=[45,45])

radius - gives the radius of the spheres(default=0.7)

color - gives the color of the spheres(default=Yellow)

light - gives the coordinates of the light bulb(default=[100,-300,200])

camera - gives the coordinates of the camera(default=[0,-40,0])

animation - sets the animation option(values: yes or no, default=no)

animationname - gives the name of the animation .ini file(default=test.ini)

frames - gives the number of intermediate frames of the animation(default=100)

precision - defines the precision of the resulting image(default=0.02)

numberofpoints - defines how many spheres will be drawn when the knot, curve

or surface type is choosed

morphing - sets the morphing option, it automatically set the animation option to

yes(values: yes or no, default=no)

ribbon - sets the ribbon option(values: yes or no, default=no)

twist - sets the number of twist if the ribbon option is set to yes(default=0);

append - set the append option, if set to yes, the image will be appended to the

file(default=no)

preview - preview the image in maple graphics(values: yes or no, default=no)

ribbonprecision - gives the precision of the ribbon(default=4)

scene - sets the scene number(default=13, values ranging from 0 to 17)

background - gives a background color, no plane or sky exist no

			 more(default=Blue)

povname - gives the name of the pov output file(extension .pov, default=test.pov)

ribbonstyle - gives the style of the ribbon(1 - ribbon, 2 - wounding curve, 3 - two

			 half ribbon with different colors, default=1)

ribboncolor - gives the color of the ribbon(default=color)

animationstyle - gives the animation styles(1 - twist around the z axis, 2 - animate

			 on a specific variable, 3 - both 1 and 2)

animationvariable - gives the specific variable, and then a list of beginning and

ending values (default=[`A`,0,2*Pi])

ribbondistance - gives the width of the ribbon(default=1)

cyclicanimation - if the animation cycles, then it should be yes (default=no);

translate - gives the overall translation of the object (default=[0,0,0]);

scale - gives the overall scaling of the object (default=[1,1,1]);

blackwhite - color / black & white (value: yes or no, default=no)

finalclock - sets the final clock value for animation(default=1)

animationline - gives a personal animation line(default=nothing)