aarjaneiro/planetmath huggingface dataset:
fundamental theorem of space curves

fundamental theorem of space curves

Informal summary.

The

curvature

and

torsion

of a

space
curve

are

invariant

with respect to Euclidean motions.

Conversely

, a
given space curve is determined up to a Euclidean motion, by its
curvature and torsion, expressed as

functions

of the arclength.

Theorem.

Let

𝜸

:

I

→

ℝ

be a

regular

, parameterized space curve, without

points of inflection

. Let

κ

⁢

(

t

)

,

τ

⁢

(

t

)

be the
corresponding curvature and torsion functions. Let

T

:

ℝ

3

→

ℝ

3

be a

Euclidean

isometry. The curvature and
torsion of the transformed curve

T

⁢

(

𝜸

⁢

(

t

)

)

are given by

κ

⁢

(

t

)

and

τ

⁢

(

t

)

, respectively.

Conversely, let

κ

,

τ

:

I

→

ℝ

be

continuous functions

,
defined on an

interval

I

⊂

ℝ

, and suppose that

κ

⁢

(

t

)

never

vanishes

. Then, there exists an arclength parameterization

𝜸

:

I

→

ℝ

of a regular, oriented space curve, without points of
inflection, such that

κ

⁢

(

t

)

and

τ

⁢

(

t

)

are the corresponding
curvature and torsion functions. If

𝜸

^

:

I

→

ℝ

is another
such space curve, then there exists a Euclidean isometry

T

:

ℝ

3

→

ℝ

3

such that

𝜸

^

⁢

(

t

)

=

T

⁢

(

𝜸

⁢

(

t

)

)

.

Title

fundamental theorem of space curves

Canonical name

FundamentalTheoremOfSpaceCurves

Date of creation

2013-03-22 13:23:28

Last modified on

2013-03-22 13:23:28

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Theorem

Classification

msc 53A04

Related topic

SpaceCurve

Generated on Fri Feb 9 22:20:43 2018 by

LaTeXML
line in plane

line in plane

Equation of a line

Suppose

a

,

b

,

c

∈

ℝ

. Then the set of points

(

x

,

y

)

in the
plane that satisfy

a

⁢

x

+

b

⁢

y

+

c

=

0

,

where

a

and

b

can not be both 0, is an (infinite)

line

.

The value of

y

when

x

=

0

, if it exists, is called the

y

-intercept

. Geometrically, if

d

is the

y

-intercept, then

(

0

,

d

)

is the point of intersection of the line and the

y

-axis. The

y

-intercept exists iff the line is not

parallel

to the

y

-axis. The

x

-intercept

is defined similarly.

If

b

≠

0

, then the above equation of the line can be rewritten as

y

=

m

⁢

x

+

d

.

This is called the

slope-intercept form

of a line, because both the slope and the

y

-intercept are easily identifiable in the equation. The slope is

m

and the

y

-intercept is

d

.

Three finite points

(

x

1

,

y

1

)

,

(

x

2

,

y

2

)

,

(

x

3

,

y

3

)

in

ℝ

2

are

collinear

if and only if the following

determinant

vanishes:

|

x

1

x

2

x

3

y

1

y

2

y

3

1

1

1

|

=

0

.

Therefore, the line through distinct points

(

x

1

,

y

1

)

and

(

x

2

,

y

2

)

has equation

|

x

1

x

2

x

y

1

y

2

y

1

1

1

|

=

0

,

or more simply

(

y

1

-

y

2

)

⁢

x

+

(

x

2

-

x

1

)

⁢

y

+

y

2

⁢

x

1

-

x

2

⁢

y

1

=

0

.

Line segment

Let

p

1

=

(

x

1

,

y

1

)

and

p

2

=

(

x

2

,

y

2

)

be distinct points in

ℝ

2

. The closed line segement generated by these points is the set

{

p

∈

ℝ

2

∣

p

=

t

⁢

p

1

+

(

1

-

t

)

⁢

p

2

,

0

≤

t

≤

1

}

.

Title

line in plane

Canonical name

LineInPlane

Date of creation

2013-03-22 15:18:29

Last modified on

2013-03-22 15:18:29

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

17

Author

matte (1858)

Entry type

Definition

Classification

msc 53A04

Classification

msc 51N20

Synonym

y-intercept

Synonym

x-intercept

Related topic

LineSegment

Related topic

SlopeAngle

Related topic

LineInSpace

Related topic

Slope

Related topic

AnalyticGeometry

Related topic

FanOfLines

Related topic

PencilOfConics

Defines

y

-intercept

Defines

x

-intercept

Defines

slope-intercept form

Generated on Fri Feb 9 22:21:12 2018 by

LaTeXML
line through an intersection point

line through an intersection point

Suppose that the lines

A

⁢

x

+

B

⁢

y

+

C

=

0

⁢

and

⁢

A

′

⁢

x

+

B

′

⁢

y

+

C

′

=

0

(1)

have an intersection point. Then for any real value of

k

, the equation

A

⁢

x

+

B

⁢

y

+

C

+

k

⁢

(

A

′

⁢

x

+

B

′

⁢

y

+

C

′

)

=

0

(2)

represents a line

passing through

that point.

In fact, the of the equation (2) is 1, and therefore it represents a line; secondly, (2) is satisfied if both equations (1) are satisfied, and therefore the line passes through that intersection point.

Example.

Determine the equation of the line passing through the point

(

-

5

,

2

)

and the intersection point of the lines

6

⁢

x

-

7

⁢

y

+

9

=

0

and

5

⁢

x

+

9

⁢

y

-

3

=

0

.

The equation of a line through the common point of those lines is

6

⁢

x

-

7

⁢

y

+

9

+

k

⁢

(

5

⁢

x

+

9

⁢

y

-

3

)

=

0

.

(3)

We have to find such a value for

k

that also

(

-

5

,

2

)

lies on the line, i.e. that the equation (3) is satisfied by the values

x

=

-

5

,

y

=

2

. So we get for determining

k

the equation

-

35

-

10

⁢

k

=

0

,

whence

k

=

-

7

2

. Using this value in (3), multiplying the equation by 2 and simplifying, we obtain the sought equation

23

⁢

x

+

77

⁢

y

-

39

=

0

.

This result would be obtained, of course, by first calculating the intersection point of the two given lines (it is

(

-

60

89

,

63

89

)

) and then forming the equation of the line passing this point and the point

(

-

5

,

2

)

, but then the calculations would have been substantially longer.

Note.

It is apparent that no value of

k

allows the equation (2) to the line

A

′

⁢

x

+

B

′

⁢

y

+

C

′

=

0

itself. Thus, if we had in the example instead the point

(

-

5

,

2

)

e.g. the point

(

6

,

-

3

)

of the line

5

⁢

x

+

9

⁢

y

-

3

=

0

, then we had the condition

66

+

0

⁢

k

=

0

which gives no value of

k

.

References

1

K. Väisälä:

Algebran oppi- ja esimerkkikirja II

. Neljäs painos.   Werner Söderström osakeyhtiö, Porvoo & Helsinki (1956).

Title

line through an intersection point

Canonical name

LineThroughAnIntersectionPoint

Date of creation

2013-03-22 17:30:28

Last modified on

2013-03-22 17:30:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Topic

Classification

msc 53A04

Classification

msc 51N20

Generated on Fri Feb 9 22:21:17 2018 by

LaTeXML
mean curvature (plane curve)

mean curvature (plane curve)

Let

Γ

be a

piecewise

C

1

planar curve

.

The

total curvature

,

κ

t

⁢

o

⁢

t

⁢

a

⁢

l

, of

Γ

is defined to be

∫

Γ

|

κ

⁢

(

s

)

|

⁢

𝑑

s

where

Γ

is parameterized by arclength

s

and

κ

⁢

(

s

)

is the

curvature

(

http://planetmath.org/CurvatureOfACurve

)
of

Γ

.

The

mean curvature

of

Γ

is defined to be the ratio of the total curvature to the length of

Γ

:

M

⁢

(

Γ

)

=

κ

t

⁢

o

⁢

t

⁢

a

⁢

l

⁢

(

Γ

)

L

⁢

(

Γ

)

Title

mean curvature (

plane curve

)

Canonical name

MeanCurvatureplaneCurve

Date of creation

2013-03-22 15:31:16

Last modified on

2013-03-22 15:31:16

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

11

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 53A04

Related topic

MeanCurvatureAtSurfacePoint

Defines

total curvature

Defines

mean curvature

Generated on Fri Feb 9 22:21:19 2018 by

LaTeXML
moving frame

moving frame

Let

M

be a

smooth manifold

. A

moving frame

(sometimes just a

frame

) on

M

is a choice, for every

P

∈

M

, of a basis for the

tangent space

T

p

⁢

M

to

M

at

P

. More formally (and abstractly), a frame is a (smooth)

section

of the

principal bundle

for

GL

n

over

M

.

Examples and Remarks

•

If

M

=

ℝ

n

, then any basis of

ℝ

n

trivially gives a frame as well.

•

A more interesting example (and perhaps a source for the definition) is when

M

=

ℝ

2

-

{

(

0

,

0

)

}

,

and we take the vectors

∂

∂

⁡

r

and

∂

∂

⁡

θ

at a point

(

r

,

θ

)

. Note that this frame cannot be extended to a smooth frame on all of

ℝ

2

.

•

Similar to the previous example, one can show that the 2-sphere admits no frames. A

manifold

which admits a (global) frame is called

parallelizable

.

•

A key example of a frame is the Frenet frame.

•

One places adjective in front of ”moving frame” if that adjective pertains to each basis, e.g. an

orthogonal frame

is a frame for which each basis is orthogonal (with respect to a given inner product). Given any frame, one can always ”orthonormalize” it in a smooth manner to provide an orthonormal frame.

•

Frames arise in general relativity as a formalization of the observation that there is no “preferred” observer standpoint.

References

1

Wikipedia’s

http://en.wikipedia.org/wiki/Moving_frame

entry on moving frame

Title

moving frame

Canonical name

MovingFrame

Date of creation

2013-03-22 16:27:01

Last modified on

2013-03-22 16:27:01

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 53A04

Synonym

frame

Related topic

TNBFrame

Defines

frame

Defines

orthonormal frame

Defines

parallelizable

Generated on Fri Feb 9 22:21:21 2018 by

LaTeXML
order of contact

order of contact

Suppose that

A

and

B

are smooth curves in

ℝ

n

which

pass through

a common point

P

. We say that

A

and

B

have zeroth order contact if their

tangents

at

P

are distinct.

Suppose that

A

and

B

are tangent at

P

. We may then set up a

coordinate
system

in which

P

is the origin and the

x

1

axis is tangent to both curves.
By the

implicit function theorem

, there will be a

neighborhood

of

P

such that

A

can be described parametrically as

x

i

=

f

i

⁢

(

x

1

)

with

i

=

2

,

…

,

n

and

B

can be described parametrically as

x

i

=

g

i

⁢

(

x

1

)

with

i

=

2

,

…

,

n

. We then define the

order of contact

of

A

and

B

at

P

to be the largest integer

m

such that all

partial derivatives

of

f

i

and

g

i

of order not

greater

than

m

at

P

are equal.

Title

order of contact

Canonical name

OrderOfContact

Date of creation

2013-03-22 16:59:49

Last modified on

2013-03-22 16:59:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53A04

Synonym

order contact

Generated on Fri Feb 9 22:21:22 2018 by

LaTeXML
polar curve

polar curve

Polar curves

are

plane curves

in

ℝ

2

that are expressed in

polar coordinates

(

r

,

θ

)

. The two simplest polar curves are obtained when one of the two

coordinates

is set to be a constant. If the first coordinate is set to a constant

r

, we have a circle with radius

|

r

|

, or a point when

r

=

0

. When the second coordinate is the constant instead, say

c

, we have a

straight line

through the (polar) origin, with slope =

tan

⁡

c

.

Using polar coordinates, one can generate many visually pleasing
curves. Below are some of the most popular ones.

Title

polar curve

Canonical name

PolarCurve

Date of creation

2013-03-22 15:17:08

Last modified on

2013-03-22 15:17:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Example

Classification

msc 53A04

Classification

msc 51-01

Synonym

limaçon

Related topic

AreaOfPlaneRegion

Related topic

CassiniOval

Defines

lemniscate

Defines

rhodonea

Defines

cardioid

Defines

limacon

Generated on Fri Feb 9 22:21:43 2018 by

LaTeXML
Serret-Frenet equations in ℝ^2

Serret-Frenet equations in

ℝ

2

Given a

plane curve

, we may associate to each point on the curve an

orthonormal basis

consisting of the

unit normal

tangent vector

and
the unit normal. In general, different points will have different
bases associated to them, so we may ask how the basis depends upon
the choice of point. The

Serret-Frenet equations

answer this
question by relating the rte of change of the basis vectors to
the

curvature

of the curve.

Suppose

I

is an

open interval

and

c

:

I

→

ℝ

2

is a twice

continuously differentiable

curve such that

∥

c

′

∥

=

1

.
Let us then
define the

tangent vector

and

normal vector

as

𝐓

=

c

′

,

𝐍

=

J

⋅

𝐓

,

where

J

=

(

0

-

1

1

0

)

is the

rotational matrix

that rotates the plane

90

degrees counterclockwise.

Curvature

Differentiating

⟨

c

′

,

c

′

⟩

=

1

yields

⟨

𝐓

′

,

𝐓

⟩

=

0

,
so

𝐓

′

is in the

orthogonal complement

of

𝐓

,
which is

1

-dimensional. Since

J

⋅

𝐓

is also in
the orthogonal complement,
it follows that there exists a

function

κ

:

I

→

ℝ

such that

𝐓

′

=

κ

⁢

J

⋅

𝐓

.

Furthermore,

κ

is uniquely determined by this equation.
We define this unique

κ

to
be the

curvature

of

c

. Explicitly,

κ

=

⟨

𝐓

′

,

J

⋅

𝐓

⟩

.

Serret-Frenet equations

By the definition of curvature

𝐓

′

=

κ

⁢

J

⋅

𝐓

=

κ

⁢

𝐍

and so

𝐍

′

=

J

⋅

𝐓

′

=

κ

⁢

J

⁢

𝐍

=

-

κ

⁢

𝐓

since

J

2

=

-

I

. These are the

Serret-Frenet

equations

(

𝐓

𝐍

)

′

=

(

0

κ

-

κ

0

)

⁢

(

𝐓

𝐍

)

.

Title

Serret-Frenet equations in

ℝ

2

Canonical name

SerretFrenetEquationsInmathbbR2

Date of creation

2013-03-22 15:16:57

Last modified on

2013-03-22 15:16:57

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53A04

Related topic

SerretFrenetFormulas

Generated on Fri Feb 9 22:22:27 2018 by

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sinusoid

sinusoid

A

sinusoid

is a curve of the form

ℝ

→

ℝ

2

t

↦

(

t

,

sin

⁡

k

⁢

t

)

,

where

k

>

0

is a

parameter

determining the

oscillation

.

The basic sinusoid, the curve

y

=

sin

⁡

x

in the

x

⁢

y

-plane, oscillates periodically with the period of sine (

http://planetmath.org/ComplexSineAndCosine

),

2

⁢

π

, as

x

increases.

•

On the

interval

[

0

,

π

2

]

,  the curve is ascending because the

derivative

of sine (

http://planetmath.org/ComplexSineAndCosine

),

cos

⁡

x

, is

positive

for

acute angles

x

.

•

Consequently, on the interval

[

π

2

,

π

]

, the

supplement formula

sin

⁡

(

π

-

x

)

=

sin

⁡

x

tells that the sinusoid is descending.

•

Thus we get on the whole interval

[

0

,

π

]

a cap-formed (

⌢

) arc.

•

Because sine is an

odd function

(

http://planetmath.org/ComplexSineAndCosine

), we have on the interval

[

-

π

,

0

]

the of the cap, a cup-formed (

⌣

) arc.

•

All in all, on the period interval

[

-

π

,

π

]

the sinusoid consists of the consecutive cup and cap, together a lying-S formed (

∽

) arc.

•

The same is repeated on each other period interval

[

(

2

⁢

n

-

1

)

⁢

π

,

(

2

⁢

n

+

1

)

⁢

π

]

where

n

∈

ℤ

.

Title

sinusoid

Canonical

name

Sinusoid

Date of creation

2015-02-04 11:23:26

Last modified on

2015-02-04 11:23:26

Owner

matte (1858)

Last modified by

pahio (2872)

Numerical id

12

Author

matte (2872)

Entry type

Definition

Classification

msc 53A04

Related topic

Trigonometry

Related topic

DefinitionsInTrigonometry

Generated on Fri Feb 9 22:22:41 2018 by

LaTeXML
slope angle

slope angle

The

slope angle

α

of any line in

ℝ

2

is the angle between (

http://planetmath.org/AngleBetweenTwoLines

) the line and

x

-axis; in the case that the line is descending, the angle is

negative

. We always have that

-

90

∘

<

α

≤

90

∘

.

For all non-vertical lines having a slope of

m

, the slope angle is

α

=

arctan

⁡

m

,

and we also have that

m

=

tan

⁡

α

.

Below are some examples of slope angles of lines that intersect the

x

-axis.

In some mathematical works, the slope angle of a line in

ℝ

2

is defined to be the angle measured from the positive

x

-axis to the line in the counterclockwise direction. This

definition

has some drawbacks because slope angles are then allowed to be obtuse but can no longer be negative. When this definition is used, the

α

=

arctan

⁡

m

no longer holds. The convention in PlanetMath is to use the former definition.

References

1

“Slope,

Distance

and Equation

Calculator

.”

1729 Software Systems.

Accessed on 24 June 2007.
URL:

http://www.1728.com/distance.htm

http://www.1728.com/distance.htm

2

“Slope angle.”

McGraw-Hill Dictionary of Scientific and Technical Terms.

McGraw-Hill Companies, Inc., 2003. Accessed via Answers.com on 7 June 2007. URL:

http://www.answers.com/topic/slope-angle

http://www.answers.com/topic/slope-angle

Title

slope angle

Canonical

name

SlopeAngle

Date of creation

2013-03-22 17:13:14

Last modified on

2013-03-22 17:13:14

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

22

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 53A04

Classification

msc 51N20

Related topic

Slope

Related topic

LineInThePlane

Related topic

ZeroesOfDerivativeOfComplexPolynomial

Generated on Fri Feb 9 22:22:44 2018 by

LaTeXML
tangent of hyperbola

tangent of hyperbola

Let us derive the equation of the

tangent line

of the

hyperbola

x

2

a

2

-

y

2

b

2

=

1

(1)

having

(

x

0

,

y

0

)

as the tangency point (

y

0

≠

0

).

If

(

x

1

,

y

1

)

is another point of the hyperbola (

x

1

≠

x

0

), the

secant line

through both points is

y

-

y

0

=

y

1

-

y

0

x

1

-

x

0

⁢

(

x

-

x

0

)

.

(2)

Since both points satisfy the equation (1) of the hyperbola, we have

0

=

1

-

1

=

(

x

1

2

a

2

-

y

1

2

b

2

)

-

(

x

0

2

a

2

-

y

0

2

b

2

)

=

(

x

1

-

x

0

)

⁢

(

x

1

+

x

0

)

a

2

-

(

y

1

-

y

0

)

⁢

(

y

1

+

y

0

)

b

2

,

which implies the proportion equation

y

1

-

y

0

x

1

-

x

0

=

b

2

⁢

(

x

1

+

x

0

)

a

2

⁢

(

y

1

+

y

0

)

.

Thus the equation (2) may be written

y

-

y

0

=

b

2

⁢

(

x

1

+

x

0

)

a

2

⁢

(

y

1

+

y

0

)

⁢

(

x

-

x

0

)

.

(3)

When we let here

x

1

→

x

0

,

y

1

→

y

0

,  this changes to the equation of the

tangent

:

y

-

y

0

=

b

2

⁢

x

0

a

2

⁢

y

0

⁢

(

x

-

x

0

)

.

(4)

A little simplification allows to write it as

x

0

⁢

x

a

2

-

y

0

⁢

y

b

2

=

x

0

2

a

2

-

y

0

2

b

2

,

i.e.

x

0

⁢

x

a

2

-

y

0

⁢

y

b

2

=

1

.

(5)

Limiting position of tangent

Putting first

y

:=

0

and then

x

:=

0

into (5) one obtains the values

x

=

a

2

x

0

and

y

=

-

b

2

y

0

on which the tangent line intersects the

coordinate

axes.  From these one sees that when the point of tangency unlimitedly moves away from the origin (

x

0

→

∞

,

y

0

→

∞

), both

intersection

points tend to the origin.  At the same time, the slope

b

2

⁢

x

0

a

2

⁢

y

0

tends to a certain limit

b

a

, because

y

0

x

0

=

b

a

⁢

x

0

2

-

a

2

:

x

0

=

b

a

⁢

1

-

a

2

x

0

2

⟶

b

a

.

Thus one infers that the limiting position of the tangent line is the

asymptote

(

http://planetmath.org/Hyperbola2

)

y

=

b

a

⁢

x

of the hyperbola.

Consequently, one can say the asymptotes of a hyperbola to be whose tangency points are infinitely far.

The tangent (5) halves the angle between the focal radii of the hyperbola drawn from

(

x

0

,

y

0

)

.

Title

tangent of hyperbola

Canonical name

TangentOfHyperbola

Date of creation

2013-03-22 19:10:31

Last modified on

2013-03-22 19:10:31

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Derivation

Classification

msc 53A04

Classification

msc 51N20

Classification

msc 51-00

Related topic

Slope

Related topic

TangentOfConicSection

Generated on Fri Feb 9 22:22:50 2018 by

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area of surface of revolution

area of surface of revolution

A

surface of revolution

is a 3D

surface

, generated when an arc is rotated fully around a

straight line

.

The general surface of revolution is obtained when the arc is rotated about an arbitrary axis. If one chooses

Cartesian coordinates

, and specializes to the case of a surface of revolution generated by rotating about the

x

-axis a curve described by

y

in the

interval

[

a

,

b

]

, its area can be calculated by the formula

A

=

2

⁢

π

⁢

∫

a

b

y

⁢

1

+

(

d

⁢

y

d

⁢

x

)

2

⁢

𝑑

x

Similarly, if the curve is rotated about the

y

-axis rather than the

x

-axis, one has the following formula:

A

=

2

⁢

π

⁢

∫

a

b

x

⁢

1

+

(

d

⁢

x

d

⁢

y

)

2

⁢

𝑑

y

The general formula is most often seen with parametric

coordinates

. If

x

⁢

(

t

)

and

y

⁢

(

t

)

describe the curve, and

x

⁢

(

t

)

is always positive or zero, then the area of the general surface of revolution

A

in the interval

[

a

,

b

]

can be calulated by the formula

A

=

2

⁢

π

⁢

∫

a

b

y

⁢

(

d

⁢

x

d

⁢

t

)

2

+

(

d

⁢

y

d

⁢

t

)

2

⁢

𝑑

t

To obtain a specific surface of revolution,

translation

or

rotation

can be used to move an arc before revolving it around an axis. For example, the specific surface of revolution around the line

y

=

s

can be found by replacing

y

with

y

-

s

, moving the arc towards the

x

-axis so

y

=

s

lies on it. Now, the surface of revolution can be found using one of the formulae above.

In this specific case, replacing

y

with

y

=

s

,  the area of a surface of revolution is found using the formula

A

=

2

⁢

π

⁢

∫

a

b

(

y

-

s

)

⁢

(

d

⁢

y

d

⁢

x

)

2

⁢

𝑑

y

Title

area of surface of revolution

Canonical name

AreaOfSurfaceOfRevolution

Date of creation

2014-07-24 18:36:37

Last modified on

2014-07-24 18:36:37

Owner

rspuzio (6075)

Last modified by

pahio (2872)

Numerical id

13

Author

rspuzio (2872)

Entry type

Topic

Classification

msc 53A05

Classification

msc 26B15

Synonym

area of revolution

Synonym

surface area of revolution

Related topic

SurfaceOfRevolution2

Related topic

VolumeOfSolidOfRevolution

Generated on Fri Feb 9 22:23:06 2018 by

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