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Like other ruled surfaces conoids are of high interest with architects, because they can be built using beams or bars. Right conoids can be manufactured easily: one threads bars onto an axis such that they can be rotated around this axis, only. Afterwards one deflects the bars by a directrix and
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1187:. The conoid has a parabola as directrix, the y-axis as axis and a plane parallel to the x-z-plane as directrix plane. It is used by architects as roof surface (s. below).
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describes a right circular conoid with the unit circle of the x-y-plane as directrix and a directrix plane, which is parallel to the y--z-plane. Its axis is the line
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551:{\displaystyle \mathbf {x} (u,v)=(\cos u,\sin u,0)+v(0,-\sin u,z_{0})\ ,\ 0\leq u<2\pi ,v\in \mathbb {R} }
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is an implicit representation. Hence the right circular conoid is a surface of degree 4.
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1018:{\displaystyle \mathbf {x} (u,v)=\left(1,u,-u^{2}\right)+v\left(-1,0,u^{2}\right)}
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353:{\displaystyle \det(\mathbf {r} ,\mathbf {\dot {r}} ,\mathbf {\ddot {r}} )=0}
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There are a lot of conoids with singular points, which are investigated in
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to its directrix plane. Hence all rulings are perpendicular to the axis.
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1128:{\displaystyle =\left(1-v,u,-(1-v)u^{2}\right)\ ,u,v\in \mathbb {R} \ ,}
16:
Ruled surface made of lines parallel to a plane and intersecting an axis
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are all parallel to the directrix plane. The planarity of the vectors
98:
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Modern differential geometry of curves and surfaces with
Mathematica
209:{\displaystyle \mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)\ }
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The implicit representation is fulfilled by the points of the line
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85:, whose rulings (lines) fulfill the additional conditions:
732:{\displaystyle (1-x^{2})(z-z_{0})^{2}-y^{2}z_{0}^{2}=0}
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The intersection with a horizontal plane is an ellipse.
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If the directrix is a circle, the conoid is called a
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618:{\displaystyle (x,0,z_{0})\ x\in \mathbb {R} \ .}
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745:gives for a right circular conoid with radius
1190:The parabolic conoid has no singular points.
1340:, 3rd ed. Boca Raton, FL:CRC Press, 2006.
1356:Geometry of curves and surfaces with MAPLE
1297:generates a conoid (s. parabolic conoid).
826:{\displaystyle V={\tfrac {\pi }{2}}r^{2}h}
136:and can be represented parametrically by
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902:parabolic conoid: directrix is a parabola
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109:All rulings intersect a fixed line, the
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19:For the organelle called conoid used by
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45: Axis is perpendicular to the
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1336:A. Gray, E. Abbena, S. Salamon,
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906:The parametric representation
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398:The parametric representation
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92:All rulings are parallel to a
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1318:Encyclopedia of Mathematics
875:{\displaystyle (x,0,z_{0})}
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39: Directrix is a circle
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18:
1307:mathworld: Plücker conoid
1180:{\displaystyle z=-xy^{2}}
886:. Such points are called
382:On Conoids and Spheroides
74: 'cone' and
1354:Vladimir Y. Rovenskii,
1275:conoids in architecture
290:can be represented by
34:Right circular conoid:
21:intracellular parasites
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1267:conoid in architecture
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81: 'similar') is a
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394:Right circular conoid
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237:with fixed parameter
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50: directrix plane
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785:the exact volume:
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1366:978-0-8176-4074-3
1349:978-1-58488-448-4
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778:{\displaystyle h}
758:{\displaystyle r}
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1387:Geometric shapes
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1253:Whitney umbrella
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1210:Whitney Umbrella
1194:Further examples
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1301:External links
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1241:Plücker conoid
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123:perpendicular
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83:ruled surface
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1292:Architecture
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1259:Applications
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1144:describes a
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119:right conoid
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1280:Mathematics
765:and height
128:Because of
25:myzocytosis
1376:Categories
1331:References
377:Archimedes
219:Any curve
64:(from
1323:EMS Press
1162:−
1112:∈
1073:−
1064:−
1049:−
986:−
957:−
802:π
696:−
673:−
651:−
602:∈
541:∈
532:π
520:≤
486:
480:−
450:
438:
371:The term
336:¨
321:˙
263:directrix
99:directrix
1382:Surfaces
1313:"Conoid"
1215:helicoid
888:singular
389:Examples
265:and the
58:geometry
1325:, 2001
267:vectors
1364:
1347:
1120:
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610:
596:
514:
508:
373:conoid
204:
96:, the
62:conoid
48:
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23:, see
101:plane
94:plane
78:ειδης
71:κωνος
68:
66:Greek
1362:ISBN
1345:ISBN
526:<
111:axis
631::
483:sin
447:sin
435:cos
301:det
130:(1)
107:(2)
90:(1)
56:In
1378::
1321:,
1315:,
1288:.
890:.
385:.
368:.
242:=
60:a
1368:)
1360:(
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1343:(
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1159:=
1156:z
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1116:R
1109:v
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1103:u
1100:,
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1055:,
1052:v
1046:1
1042:(
1038:=
1012:)
1006:2
1002:u
998:,
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989:1
982:(
978:v
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971:)
965:2
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954:,
951:u
948:,
945:1
941:(
937:=
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931:v
928:,
925:u
922:(
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870:)
865:0
861:z
857:,
854:0
851:,
848:x
845:(
833:.
821:h
816:2
812:r
805:2
796:=
793:V
773:h
753:r
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724:=
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714:0
710:z
704:2
700:y
691:2
687:)
681:0
677:z
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667:(
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648:1
645:(
613:.
606:R
599:x
593:)
588:0
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580:,
577:0
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571:x
568:(
545:R
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471:(
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414:(
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360:.
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327:,
318:r
312:,
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304:(
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284:(
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273:(
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255:(
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226:u
224:(
222:x
201:)
198:u
195:(
191:r
187:v
184:+
181:)
178:u
175:(
171:c
167:=
164:)
161:v
158:,
155:u
152:(
148:x
113:.
103:.
76:-
27:.
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