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Constructing ovals, ovoids and spirals
 

Constructing ovals, ovoids and spirals

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    Constructing ovals, ovoids and spirals Constructing ovals, ovoids and spirals Presentation Transcript

    • Constructing ovals,ovoids and spirals
    • Draw an oval knowing thelenght of axes AB and CD
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • Draw an arc with radius CP and the centre on C until it intersects segment AC at V. P C A O B D
    • 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
    • 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
    • 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
    • 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
    • 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
    • Draw its perpendicular bisector AV, wich willintersect OD or its extension at point M, as well as the major semiaxis at point N. P C V A O B D
    • Draw its perpendicular bisector AV, wich willintersect 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
    • Draw its perpendicular bisector AV, wich willintersect 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
    • Draw its perpendicular bisector AV, wich willintersect 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
    • Draw its perpendicular bisector AV, wich willintersect 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
    • Draw its perpendicular bisector AV, wich willintersect 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
    • Draw its perpendicular bisector AV, wich willintersect 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
    • Draw its perpendicular bisector AV, wich willintersect 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • Join points M and M’ to N and N’ and draw arcswith centres M’ and M and with radius M’D and MC. P M C V N N A O B D M
    • Join points M and M’ to N and N’ and draw arcswith centres M’ and M and with radius M’D and MC. P M C V N N N A O B D M
    • Join points M and M’ to N and N’ and draw arcswith centres M’ and M and with radius M’D and MC. P M C C V N N N A O B D M
    • Join points M and M’ to N and N’ and draw arcswith centres M’ and M and with radius M’D and MC. P M C C V N N N A O B D M
    • Join points M and M’ to N and N’ and draw arcswith centres M’ and M and with radius M’D and MC. P M C C V N N N A O B D M
    • Join points M and M’ to N and N’ and draw arcswith 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
    • Join points M and M’ to N and N’ and draw arcswith 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
    • Join points M and M’ to N and N’ and draw arcswith 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
    • Join points M and M’ to N and N’ and draw arcswith 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
    • Join points M and M’ to N and N’ and draw arcswith 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • Draw an ovoid knowing that the lenght of its minor axis is AB
    • Get point O by drawing the perpendicular bisector of the known axis AB. A B
    • Get point O by drawing the perpendicular bisector of the known axis AB. 0° 0 A B
    • Get point O by drawing the perpendicular bisector of the known axis AB. 0° 104 A B
    • Get point O by drawing the perpendicular bisector of the known axis AB. 0° B 104 A
    • Get point O by drawing the perpendicular bisector of the known axis AB. 0° 104 A B
    • Get point O by drawing the perpendicular bisector of the known axis AB. 0° 104 A B
    • Get point O by drawing the perpendicular bisector of the known axis AB. A B
    • Get point O by drawing the perpendicular bisector of the known axis AB. A B
    • Get point O by drawing the perpendicular bisector of the known axis AB. A B O
    • Get point O by drawing the perpendicular bisector of the known axis AB. A B O
    • With its centre at O and radius OA, draw acircumference that intersects the perpendicular bisector at point P. A B O
    • With its centre at O and radius OA, draw acircumference that intersects the perpendicular bisector at point P. 0° 86 A B O
    • With its centre at O and radius OA, draw acircumference that intersects the perpendicular bisector at point P. 0° 86 A B O
    • With its centre at O and radius OA, draw acircumference that intersects the perpendicular bisector at point P. A B O
    • With its centre at O and radius OA, draw acircumference that intersects the perpendicular bisector at point P. A B O P
    • Join points A and B to P to give us the lines r and s. A B O P
    • Join points A and B to P to give us the lines r and s. A B O P
    • Join points A and B to P to give us the lines r and s. A B A O B O P P
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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°
    • 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°
    • 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°
    • Draw a spiral withcentres O and O’
    • Draw a straight line and define two points on it, O and O’. O O
    • With point O as your centre, and using a radius ofthe segment OO’, draw a semi circle (which will give us point A at its intersection with the straight line). O O 0 0°
    • With point O as your centre, and using a radius ofthe segment OO’, draw a semi circle (which will give us point A at its intersection with the straight line). O O 86 0°
    • With point O as your centre, and using a radius ofthe segment OO’, draw a semi circle (which will give us point A at its intersection with the straight line). 0° 86 O O
    • With point O as your centre, and using a radius ofthe segment OO’, draw a semi circle (which will give us point A at its intersection with the straight line). O O
    • With point O as your centre, and using a radius ofthe segment OO’, draw a semi circle (which will give us point A at its intersection with the straight line). A O O
    • Using point O’ as the centre and radius O’A, drawanother semi circle to get point B on the straight line. A O O
    • Using point O’ as the centre and radius O’A, drawanother semi circle to get point B on the straight line. 0° 0 A O O
    • Using point O’ as the centre and radius O’A, drawanother semi circle to get point B on the straight line. 0° 172 A O O
    • Using point O’ as the centre and radius O’A, drawanother semi circle to get point B on the straight line. A O O 172 0°
    • Using point O’ as the centre and radius O’A, drawanother semi circle to get point B on the straight line. A O O B
    • Draw another semi circle using O as its centre andwith 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
    • Draw another semi circle using O as its centre andwith 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°
    • Draw another semi circle using O as its centre andwith 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
    • Draw another semi circle using O as its centre andwith 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
    • Draw another semi circle using O as its centre andwith 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
    • Draw a spiral withcentres on the vertices of the equilateral triangle
    • Draw an equilateral, extend its sides and number its vertices: 1, 2 and 3. 1 3 2
    • Draw an equilateral, extend its sides and number its vertices: 1, 2 and 3. 1 3 2
    • With radius 1-3 and its centre at point 1, draw an arcuntil we define point A on the extension of side 2-1 of the triangle. 1 3 2
    • With radius 1-3 and its centre at point 1, draw an arcuntil we define point A on the extension of side 2-1 of the triangle. 0° 0 1 3 2
    • With radius 1-3 and its centre at point 1, draw an arcuntil we define point A on the extension of side 2-1 of the triangle. 0° 81 1 3 2
    • With radius 1-3 and its centre at point 1, draw an arcuntil we define point A on the extension of side 2-1 of the triangle. 0° 81 1 3 2
    • With radius 1-3 and its centre at point 1, draw an arcuntil we define point A on the extension of side 2-1 of the triangle. 1 3 2
    • With radius 1-3 and its centre at point 1, draw an arcuntil we define point A on the extension of side 2-1 of the triangle. A 1 3 2
    • 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
    • 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
    • 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°
    • 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
    • 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
    • Alternate this procedure successively (centre at 3and radius 3B to define point C, etc.) to draw out the spiral. A 1 3 B 2
    • Alternate this procedure successively (centre at 3and radius 3B to define point C, etc.) to draw out the spiral. A 1 3 B 2 0 0°
    • Alternate this procedure successively (centre at 3and radius 3B to define point C, etc.) to draw out the spiral. A 1 3 B 2 249 0°
    • Alternate this procedure successively (centre at 3and radius 3B to define point C, etc.) to draw out the spiral. A 1 3 B 2 0° 249
    • Alternate this procedure successively (centre at 3and radius 3B to define point C, etc.) to draw out the spiral. A 1 3 B 2
    • Alternate this procedure successively (centre at 3and radius 3B to define point C, etc.) to draw out the spiral. A 1 3 B 2 C
    • Constructing ovals,ovoids and spirals