The document provides step-by-step instructions for constructing ovals, ovoids, and spirals. It begins by showing how to draw an oval given the lengths of its major and minor axes. It then demonstrates how to obtain the center point and draw various arcs and lines to fully construct the oval shape. Similar techniques are used to construct an ovoid given only the length of its minor axis.

1. Constructing ovals,
ovoids and spirals

3. Draw an oval knowing the
lenght of axes AB and CD

4. Draw an arc with its center at O and with radius of
OA that intersects the extension of CD (minor
axis) at point P. Join points A and C.
C
A O B
D

5. Draw an arc with its center at O and with radius of
OA that intersects the extension of CD (minor
axis) at point P. Join points A and C.
C
A O B
D

6. Draw an arc with its center at O and with radius of
OA that intersects the extension of CD (minor
axis) at point P. Join points A and C.
0°
179
C
A O B
D

7. Draw an arc with its center at O and with radius of
OA that intersects the extension of CD (minor
axis) at point P. Join points A and C.
C
179 0°
A O B
D

8. Draw an arc with its center at O and with radius of
OA that intersects the extension of CD (minor
axis) at point P. Join points A and C.
P
C
A O B
D

9. Draw an arc with its center at O and with radius of
OA that intersects the extension of CD (minor
axis) at point P. Join points A and C.
P
C
A O B
D

10. Draw an arc with radius CP and the centre on C
until it intersects segment AC at V.
P
C
A O B
D

11. Draw an arc with radius CP and the centre on C
until it intersects segment AC at V.
P
58 0°
C
A O B
D

12. Draw an arc with radius CP and the centre on C
until it intersects segment AC at V.
131°
58
P
58 0°
C
A O B
D

13. Draw an arc with radius CP and the centre on C
until it intersects segment AC at V.
131°
58
P
58 0°
C
A O B
D

14. Draw an arc with radius CP and the centre on C
until it intersects segment AC at V.
131°
58
P
58 0°
C
V
A O B
D

15. Draw an arc with radius CP and the centre on C
until it intersects segment AC at V.
131°
58
P
58 0°
C
V
A O B
D

16. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
P
C
V
A O B
D

17. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
0°
91
P
C
V
A O B
D

18. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
0°
91 0°
91
P
C
V
A O B
D

19. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
0°
91 0°
91
P
C
V
0° 91
A O B
D

20. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
0°
91 0°
91
P
C
V
0° 91
A O B
D

21. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
0°
91 0°
91
P
C
V
0° 91
A O B
D

22. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
0°
91 0°
91
P
C
V
0° 91
N
A O B
D
M

23. Draw its perpendicular bisector AV, wich will
intersect OD or its extension at point M, as well as
the major semiaxis at point N.
0°
91 0°
91
P
C
V
0° 91
N
A O B
D
M

24. Draw the symmetrical points of M and N on the
opposite ide of the axes, M’ and N’.
P
C
V
N
A O B
D
M

25. Draw the symmetrical points of M and N on the
opposite ide of the axes, M’ and N’.
P
C
V
N
A O B
140 0°
D
M

26. Draw the symmetrical points of M and N on the
opposite ide of the axes, M’ and N’.
0°
P
86
C
V
N
A O B
140 0°
D
M

27. Draw the symmetrical points of M and N on the
opposite ide of the axes, M’ and N’.
0°
P 10°
86 86
C
V
N
A O B
140 0°
D
M

28. Draw the symmetrical points of M and N on the
opposite ide of the axes, M’ and N’.
0°
P 10°
86 M' 86
C
V
N N'
A O B
140 0°
D
M

29. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
M'
C
V
N N'
A O B
D
M

30. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
M'
C
V
N N'
N'
A O B
D
M

31. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
M'
C
C
V
N N'
N'
A O B
D
M

32. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
M'
C
C
V
N N'
N'
A O B
D
M

33. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
M'
C
C
V
N N'
N'
A O B
D
M

34. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
M'
C
C
V
N N'
N'
A 0° 257
O B
D
M

35. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
P
M'
M'
C
C
V C
V
N
N
0°
257 N'
N'
N'
A
A O B
B
0° 257
O
D
D
M
M

36. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
P
M'
M'
C
C
V C
V
N
N
0°
257 257 0° N'
N'
N'
A
A O B
B
0° 257
O
D
D
M
M

37. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
P
M'
M'
C
C
V C
V 257
0°
N
N
0°
257 257 0° N'
N'
N'
A
A O B
B
0° 257
O
D
D
M
M

38. Join points M and M’ to N and N’ and draw arcs
with centres M’ and M and with radius M’D and MC.
P
P
M'
M'
C
C
V C
V 257
0°
N
N
0°
257 257 0° N'
N'
N'
A
A O B
B
0° 257
O
D
D
M
M

39. Finally, draw arcs with centres N and N’ and radius
NA and N’B.
P
M'
C
V
N N'
A O B
D
M

40. Finally, draw arcs with centres N and N’ and radius
NA and N’B.
0°
P
95 M'
C
V
N N'
A O B
D
M

41. Finally, draw arcs with centres N and N’ and radius
NA and N’B.
0°
P
95 M' 0°
95
C
V
N N'
A O B
D
M

42. Finally, draw arcs with centres N and N’ and radius
NA and N’B.
0°
P 0°
95 M' 0°
95
95
C
V
N N'
A O B
D
M

43. Finally, draw arcs with centres N and N’ and radius
NA and N’B.
0°
P 0°
95 M' 0°
95
95
C
V
N N'
A O B
D
M

44. Finally, draw arcs with centres N and N’ and radius
NA and N’B.
0°
P 0°
95 M' 0° 0°
95
9595
C
V
N N'
A O B
D
M

45. Finally, draw arcs with centres N and N’ and radius
NA and N’B.
0°
P 0°
95 M' 0° 0°
95
9595
C
V
N N'
A O B
D
M

46. Draw an ovoid knowing
that the lenght of its
minor axis is AB

47. Get point O by drawing the perpendicular bisector
of the known axis AB.
A B

48. Get point O by drawing the perpendicular bisector
of the known axis AB.
0°
0
A B

49. Get point O by drawing the perpendicular bisector
of the known axis AB.
0°
104
A B

50. Get point O by drawing the perpendicular bisector
of the known axis AB.
0°
B
104
A

51. Get point O by drawing the perpendicular bisector
of the known axis AB.
0°
104
A B

52. Get point O by drawing the perpendicular bisector
of the known axis AB.
0°
104
A B

53. Get point O by drawing the perpendicular bisector
of the known axis AB.
A B

54. Get point O by drawing the perpendicular bisector
of the known axis AB.
A B

55. Get point O by drawing the perpendicular bisector
of the known axis AB.
A B
O

56. Get point O by drawing the perpendicular bisector
of the known axis AB.
A B
O

57. With its centre at O and radius OA, draw a
circumference that intersects the perpendicular
bisector at point P.
A B
O

58. With its centre at O and radius OA, draw a
circumference that intersects the perpendicular
bisector at point P.
0°
86
A B
O

59. With its centre at O and radius OA, draw a
circumference that intersects the perpendicular
bisector at point P.
0°
86
A B
O

60. With its centre at O and radius OA, draw a
circumference that intersects the perpendicular
bisector at point P.
A B
O

61. With its centre at O and radius OA, draw a
circumference that intersects the perpendicular
bisector at point P.
A B
O
P

62. Join points A and B to P to give us the lines r and s.
A B
O
P

63. Join points A and B to P to give us the lines r and s.
A B
O
P

64. Join points A and B to P to give us the lines r and s.
A B
A O B
O
P
P

65. Join points A and B to P to give us the lines r and s.
A B
A O B
O
P
P
r s

66. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
A B
O
P
r s

67. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
0°
172
AA BB
O
O
PP
rr ss

68. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
0°
172
AA BB
O
O
PP
rr ss

69. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
0°
0°
172
172
AA BB
O
O
PP
rr ss

70. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
0°
0°
172
172
AA BB
O
O
PP
rr ss

71. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
0°
0°
172
172
AA BB
O
O
PP
rr ss

72. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
0°
0°
172
172
AA BB
O
O
PP
rr ss

73. Draw two arcs with radius AB and centres at points
A and B to get the points M and M’.
0°
0°
172
172
AA BB
O
O
PP
M M'
rr ss

74. With P as its centre and radius PM or PM’, draw the
last arc to finish our ovoid.
A B
O
P
M M'
r s

75. With P as its centre and radius PM or PM’, draw the
last arc to finish our ovoid.
0°
0
AA BB
OO
PP
M
M M'
M'
rr ss

76. With P as its centre and radius PM or PM’, draw the
last arc to finish our ovoid.
0°
0
0°
53
AA BB
OO
PP
M
M M'
M'
rr ss

77. With P as its centre and radius PM or PM’, draw the
last arc to finish our ovoid.
0°
0
0°
53
AA BB
OO
PP
M
M M'
M'
rr ss
53
0°

78. With P as its centre and radius PM or PM’, draw the
last arc to finish our ovoid.
0°
0
0°
53
AA BB
OO
PP
M
M M'
M'
rr ss
53
0°

79. With P as its centre and radius PM or PM’, draw the
last arc to finish our ovoid.
0°
0
0°
53
AA BB
O
O
PP
M
M M'
M'
rr ss
53
0°

80. Draw a spiral with
centres O and O’

81. Draw a straight line and define two points on it, O
and O’.
O O'

82. With point O as your centre, and using a radius of
the segment OO’, draw a semi circle (which will give
us point A at its intersection with the straight line).
O O'
0
0°

83. With point O as your centre, and using a radius of
the segment OO’, draw a semi circle (which will give
us point A at its intersection with the straight line).
O O'
86
0°

84. With point O as your centre, and using a radius of
the segment OO’, draw a semi circle (which will give
us point A at its intersection with the straight line).
0°
86
O O'

85. With point O as your centre, and using a radius of
the segment OO’, draw a semi circle (which will give
us point A at its intersection with the straight line).
O O'

86. With point O as your centre, and using a radius of
the segment OO’, draw a semi circle (which will give
us point A at its intersection with the straight line).
A O O'

87. Using point O’ as the centre and radius O’A, draw
another semi circle to get point B on the straight line.
A O O'

88. Using point O’ as the centre and radius O’A, draw
another semi circle to get point B on the straight line.
0°
0
A O O'

89. Using point O’ as the centre and radius O’A, draw
another semi circle to get point B on the straight line.
0°
172
A O O'

90. Using point O’ as the centre and radius O’A, draw
another semi circle to get point B on the straight line.
A O O'
172
0°

91. Using point O’ as the centre and radius O’A, draw
another semi circle to get point B on the straight line.
A O O' B

92. Draw another semi circle using O as its centre and
with a radius OB to get point C on the straight line.
And so, in this way, we will get our spiral.
A O O' B

93. Draw another semi circle using O as its centre and
with a radius OB to get point C on the straight line.
And so, in this way, we will get our spiral.
A O O' B
257
0°

94. Draw another semi circle using O as its centre and
with a radius OB to get point C on the straight line.
And so, in this way, we will get our spiral.
0°
257
A O O' B

95. Draw another semi circle using O as its centre and
with a radius OB to get point C on the straight line.
And so, in this way, we will get our spiral.
A O O' B

96. Draw another semi circle using O as its centre and
with a radius OB to get point C on the straight line.
And so, in this way, we will get our spiral.
C A O O' B

97. Draw a spiral with
centres on the vertices of
the equilateral triangle

98. Draw an equilateral, extend its sides and number its
vertices: 1, 2 and 3.
1
3
2

99. Draw an equilateral, extend its sides and number its
vertices: 1, 2 and 3.
1
3
2

100. With radius 1-3 and its centre at point 1, draw an arc
until we define point A on the extension of side 2-1 of
the triangle.
1
3
2

101. With radius 1-3 and its centre at point 1, draw an arc
until we define point A on the extension of side 2-1 of
the triangle.
0°
0
1
3
2

102. With radius 1-3 and its centre at point 1, draw an arc
until we define point A on the extension of side 2-1 of
the triangle.
0°
81
1
3
2

103. With radius 1-3 and its centre at point 1, draw an arc
until we define point A on the extension of side 2-1 of
the triangle.
0°
81
1
3
2

104. With radius 1-3 and its centre at point 1, draw an arc
until we define point A on the extension of side 2-1 of
the triangle.
1
3
2

105. With radius 1-3 and its centre at point 1, draw an arc
until we define point A on the extension of side 2-1 of
the triangle.
A
1
3
2

106. Repeat this process with radius 2A and the centre at
point 2 (making an arc that give us point B on the
extension of side 3-2 of the triangle).
A
1
3
2

107. Repeat this process with radius 2A and the centre at
point 2 (making an arc that give us point B on the
extension of side 3-2 of the triangle).
A
0°
0
1
3
2

108. Repeat this process with radius 2A and the centre at
point 2 (making an arc that give us point B on the
extension of side 3-2 of the triangle).
A
1
3
2
164
0°

109. Repeat this process with radius 2A and the centre at
point 2 (making an arc that give us point B on the
extension of side 3-2 of the triangle).
A
1
3
2

110. Repeat this process with radius 2A and the centre at
point 2 (making an arc that give us point B on the
extension of side 3-2 of the triangle).
A
1
3 B
2

111. Alternate this procedure successively (centre at 3
and radius 3B to define point C, etc.) to draw out the
spiral.
A
1
3 B
2

112. Alternate this procedure successively (centre at 3
and radius 3B to define point C, etc.) to draw out the
spiral.
A
1
3 B
2
0
0°

113. Alternate this procedure successively (centre at 3
and radius 3B to define point C, etc.) to draw out the
spiral.
A
1
3 B
2
249
0°

114. Alternate this procedure successively (centre at 3
and radius 3B to define point C, etc.) to draw out the
spiral.
A
1
3 B
2
0°
249

115. Alternate this procedure successively (centre at 3
and radius 3B to define point C, etc.) to draw out the
spiral.
A
1
3 B
2

116. Alternate this procedure successively (centre at 3
and radius 3B to define point C, etc.) to draw out the
spiral.
A
1
3 B
2
C

117. Constructing ovals,
ovoids and spirals