aarjaneiro/planetmath huggingface dataset:
volume of solid of revolution

volume of solid of revolution

Let us consider a

solid of revolution

, which is generated when a planar domain

D

rotates about a line of the same plane. We chose this line for the

x

-axis, and for

simplicity

we assume that the

boundaries

of

D

are the mentioned axis, two

ordinates

x

=

a

,

x

=

b

(

>

a

)

, and a

continuous

curve

y

=

f

⁢

(

x

)

.

Between the bounds

a

anb

b

we fit a sequence of points

x

1

,

x

2

,

…

,

x

n

-

1

and draw through these the ordinates which divide the domain

D

in

n

parts. Moreover we form for every part the (maximal)

inscribed

and the (minimal)

circumscribed

rectangle

. In the revolution of

D

, each rectangle generates a circular cylinder. The considered solid of revolution is part of the volume

V

>

of the union of the cyliders

generated by

the circumscribed rectangles and at the same time contains the volume

V

<

of the union of the

cylinders

generated by the inscribed rectangles.

Now it is apparent that

V

>

=

π

⁢

[

M

1

2

⁢

(

x

1

-

a

)

+

M

2

2

⁢

(

x

2

-

x

1

)

+

…

+

M

n

2

⁢

(

b

-

x

n

-

1

)

]

,

V

<

=

π

⁢

[

m

1

2

⁢

(

x

1

-

a

)

+

m

2

2

⁢

(

x

2

-

x

1

)

+

…

+

m

n

2

⁢

(

b

-

x

n

-

1

)

]

,

where

M

1

,

M

2

,

…

,

M

n

are the greatest and

m

1

,

m

2

,

…

,

m

n

the least values of the

continuous function

f

on the

intervals

(

http://planetmath.org/Interval

)

[

a

,

x

1

]

,

[

x

1

,

x

2

]

, …,

[

x

n

-

1

,

b

]

. The volume

V

of the solid of revolution thus satisfies

V

<

≤

V

≤

V

>

,

and this is true for any

x

1

<

x

2

<

…

<

x

n

-

1

of the interval

[

a

,

b

]

.
The theory of the

Riemann integral

guarantees that there exists only one

real number

V

having this property and that it is also the definition of the

integral

∫

a

b

π

⁢

[

f

⁢

(

x

)

]

2

⁢

𝑑

x

.

Therefore the volume of the given solid of revolution can be obtained from

V

=

π

⁢

∫

a

b

[

f

⁢

(

x

)

]

2

⁢

𝑑

x

.

References

1

E. Lindelöf

:

Johdatus korkeampaan analyysiin

. Neljäs painos.  Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).

Title

volume of solid of revolution

Canonical name

VolumeOfSolidOfRevolution

Date of creation

2013-03-22 17:20:12

Last modified on

2013-03-22 17:20:12

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Topic

Classification

msc 51M25

Related topic

PappussTheoremForSurfacesOfRevolution

Related topic

SurfaceOfRevolution

Related topic

VolumeAsIntegral

Generated on Fri Feb 9 22:09:22 2018 by

LaTeXML
cevian

cevian

A

cevian

of a

triangle

, is any

line segment

joining a vertex with a point of the opposite side.

A

⁢

D

is a cevian of

△

⁢

A

⁢

B

⁢

C

.

If

D

is the

midpoint

of

B

⁢

C

, then the cevian

A

⁢

D

is a

median

. If

A

⁢

D

is

perpendicular

to

B

⁢

C

, then the cevian is a

height

.

Title

cevian

Canonical name

Cevian

Date of creation

2013-03-22 12:10:57

Last modified on

2013-03-22 12:10:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51M99

Related topic

Triangle

Related topic

CevasTheorem

Related topic

ProofOfApolloniusTheorem2

Related topic

TrigonometricVersionOfCevasTheorem

Related topic

Median

Related topic

HeightOfATriangle

Generated on Fri Feb 9 22:09:24 2018 by

LaTeXML
chordal

chordal

By the entry, the power of the point

(

a

,

b

)

with respect to the circle

K

1

⁢

(

x

,

y

)

:=

(

x

-

x

1

)

2

+

(

y

-

y

1

)

2

-

r

1

2

=

0

is equal to

K

1

⁢

(

a

,

b

)

and with respect to the circle

K

2

⁢

(

x

,

y

)

:=

(

x

-

x

2

)

2

+

(

y

-

y

2

)

2

-

r

2

2

=

0

equal to

K

2

⁢

(

a

,

b

)

.  Thus the locus of all points

(

x

,

y

)

having the same with respect to both circles is characterized by the equation

K

1

⁢

(

x

,

y

)

=

K

2

⁢

(

x

,

y

)

.

This reduces to the form

2

⁢

(

x

2

-

x

2

)

⁢

x

+

2

⁢

(

y

2

-

y

1

)

⁢

y

+

k

=

0

,

and hence the locus is a

straight line

perpendicular

to the of the circles.  This locus is called the

chordal

or the

radical axis

of the circles.

Title

chordal

Canonical name

Chordal

Date of creation

2013-03-22 15:07:50

Last modified on

2013-03-22 15:07:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Result

Classification

msc 51M99

Classification

msc 51N20

Synonym

radical axis

Generated on Fri Feb 9 22:09:26 2018 by

LaTeXML
Euler line proof

Euler line proof

Let

O

the

circumcenter

of

△

⁢

A

⁢

B

⁢

C

and

G

its

centroid

. Extend

O

⁢

G

until a point

P

such that

O

⁢

G

/

G

⁢

P

=

1

/

2

. We’ll prove that

P

is the

orthocenter

H

.

Draw the median

A

⁢

A

′

where

A

′

is the

midpoint

of

B

⁢

C

.

Triangles

O

⁢

G

⁢

A

′

and

P

⁢

G

⁢

A

are

similar

, since

G

⁢

P

=

2

⁢

G

⁢

O

,

A

⁢

G

=

2

⁢

A

′

⁢

G

and

∠

⁢

O

⁢

G

⁢

A

′

=

∠

⁢

P

⁢

G

⁢

A

. Then

∠

⁢

O

⁢

A

′

⁢

G

=

∠

⁢

P

⁢

G

⁢

A

and

O

A

′

∥

A

P

. But

O

⁢

A

′

⟂

B

⁢

C

so

A

⁢

P

⟂

B

⁢

C

, that is,

A

⁢

P

is a height of the triangle.

Repeating the same argument for the other medians proves that

P

lies on the three heights and therefore it must be the orthocenter

H

.

The ratio is

O

⁢

G

/

G

⁢

H

=

1

/

2

since we constructed it that way.

Title

Euler line proof

Canonical name

EulerLineProof

Date of creation

2013-03-22 11:44:29

Last modified on

2013-03-22 11:44:29

Owner

drini (3)

Last modified by

drini (3)

Numerical id

15

Author

drini (3)

Entry type

Proof

Classification

msc 51M99

Classification

msc 55U10

Classification

msc 18E30

Classification

msc 18-00

Classification

msc 55U35

Classification

msc 46-01

Classification

msc 47B25

Classification

msc 81-01

Related topic

EulerLine

Generated on Fri Feb 9 22:09:28 2018 by

LaTeXML
incenter

incenter

The

incenter

of a geometrical shape is the center of the

incircle

(if it has any). The radius of the incircle is sometimes
called the

inradius

.

On a

triangle

the incenter always exists and it is the

intersection

point of the three

internal angle bisectors

. So in the next picture,

A

⁢

X

,

B

⁢

Y

,

C

⁢

Z

are

angle bisectors

, and

A

⁢

B

,

B

⁢

C

,

C

⁢

A

are

tangent

to the
circle.

Title

incenter

Canonical

name

Incenter

Date of creation

2013-03-22 12:11:12

Last modified on

2013-03-22 12:11:12

Owner

mps (409)

Last modified by

mps (409)

Numerical id

11

Author

mps (409)

Entry type

Definition

Classification

msc 51M99

Synonym

incentre

Related topic

Incircle

Related topic

LengthsOfAngleBisectors

Related topic

AngleBisectorAsLocus

Related topic

Orthocenter

Related topic

Triangle

Related topic

CevasTheorem

Related topic

LemoinePoint

Related topic

GergonnePoint

Related topic

GergonneTriangle

Related topic

TrigonometricVersionOfCevasTheorem

Defines

inradius

Generated on Fri Feb 9 22:09:31 2018 by

LaTeXML
incircle

incircle

The

incircle

or

inscribed circle

of a

triangle

is a circle interior to the triangle and

tangent

to its three sides.

Moreover, the incircle of a

polygon

is an interior circle tangent to all of the polygon’s sides. Not every polygon has an inscribed circle, but triangles always do.

The center of the incircle is called the

incenter

, and it’s located at the point where the three

angle bisectors

intersect.

If the sides of a triangle are

x

,

y

and

z

, the area

A

and the

semiperimeter

p

, then the radius of incircle may be calculated from

r

=

2

⁢

A

x

+

y

+

z

=

A

p

=

(

p

-

x

)

⁢

(

p

-

y

)

⁢

(

p

-

z

)

p

.

Title

incircle

Canonical

name

Incircle

Date of creation

2013-03-22 12:11:09

Last modified on

2013-03-22 12:11:09

Owner

drini (3)

Last modified by

drini (3)

Numerical id

8

Author

drini (3)

Entry type

Definition

Classification

msc 51M99

Related topic

LemoinePoint

Related topic

Incenter

Related topic

LemoineCircle

Related topic

Triangle

Related topic

GergonnePoint

Related topic

GergonneTriangle

Related topic

ConstructionOfTangent

Generated on Fri Feb 9 22:09:34 2018 by

LaTeXML
limited tangent

limited tangent

Let the

tangent line

t

of a given curve

γ

have the tangency point

T

and go through another point

P

.  Then the segment

P

⁢

T

of

t

is a

limited tangent

from the point

P

to

γ

.

Title

limited tangent

Canonical name

LimitedTangent

Date of creation

2013-03-22 15:07:11

Last modified on

2013-03-22 15:07:11

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 51M99

Synonym

restricted tangent

Generated on Fri Feb 9 22:09:35 2018 by

LaTeXML
power of point

power of point

Theorem

.

If a secant of the circle is drawn through a point (

P

), then the

product

of the

line segments

on the secant between the point and the

perimeter

of the circle is on the direction of the secant.  The product is called

the power of the point with respect to the circle

.

Proof.

Let

P

⁢

A

and

P

⁢

B

be the segments of a secant and

P

⁢

A

′

and

P

⁢

B

′

the segments of another secant.  Then the

triangles

P

⁢

A

⁢

B

′

and

P

⁢

A

′

⁢

B

are

similar

since they have equal angles, namely the

central angles

∠

⁢

A

⁢

P

⁢

B

′

and

∠

⁢

B

⁢

P

⁢

A

′

and the

inscribed angles

∠

⁢

P

⁢

A

⁢

B

′

and

∠

⁢

P

⁢

A

′

⁢

B

.  Thus we have the proportion (

http://planetmath.org/ProportionEquation

)

P

⁢

A

P

⁢

A

′

=

P

⁢

B

′

P

⁢

B

,

which implies the asserted

equation

P

⁢

A

⋅

P

⁢

B

=

P

⁢

A

′

⋅

P

⁢

B

′

.

Notes.

If the point

P

is outside a circle, then value of the power of the point with respect to the circle is equal to the square of the

limited tangent

of the circle from

P

; this square (

http://planetmath.org/SquareOfANumber

) may be considered as the limit case of the

power of point

where the both

intersection

points of the secant with the circle coincide.  Another of the notion power of point is got when the line through

P

does not intersect the circle; we can think that then the intersecting points are

imaginary

; also now the product of the “

imaginary line

segments” is the same.

Denote by

p

2

the power of the point

P

:=

(

a

,

b

)

with respect to circle

K

⁢

(

x

,

y

)

:=

(

x

-

x

0

)

2

+

(

y

-

y

0

)

2

-

r

2

=

0

.

Then, by the

Pythagorean theorem

, we obtain

p

2

=

(

a

-

x

0

)

2

+

(

b

-

y

0

)

2

-

r

2

(1)

if

P

is outside the circle and

p

2

=

r

2

-

(

(

a

-

x

0

)

2

+

(

b

-

y

0

)

2

)

(2)

if

P

is inside of the circle.  If in the latter case, we change the definition of the power of point to be the negative value

-

p

2

for a point inside the circle, then in both cases the power of the point

(

a

,

b

)

is equal to

K

⁢

(

a

,

b

)

≡

(

a

-

x

0

)

2

+

(

b

-

y

0

)

2

-

r

2

.

Title

power of point

Canonical name

PowerOfPoint

Date of creation

2013-03-22 15:07:02

Last modified on

2013-03-22 15:07:02

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

15

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 51M99

Synonym

power of the point

Synonym

power of a point

Related topic

InversionOfPlane

Related topic

VolumeOfSphericalCapAndSphericalSector

Generated on Fri Feb 9 22:09:37 2018 by

LaTeXML
proof of triangle incenter

proof of triangle incenter

In

△

⁢

A

⁢

B

⁢

C

and construct

bisectors

of the angles at

A

and

C

, intersecting at

O

1

1

Note that the

angle bisectors

must intersect by Euclid’s Postulate 5, which states that “if a

straight line

falling on two straight lines makes the

interior angles

on the same side less than two

right angles

, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” They must meet inside the

triangle

by considering which side of

A

⁢

B

and

C

⁢

B

they fall on.

Draw

B

⁢

O

. We show that

B

⁢

O

bisects the angle at

B

, and that

O

is in fact the

incenter

of

△

⁢

A

⁢

B

⁢

C

.

Drop perpendiculars

from

O

to each of the three sides, intersecting the sides in

D

,

E

, and

F

. Clearly, by AAS,

△

⁢

C

⁢

O

⁢

D

≅

△

⁢

C

⁢

O

⁢

E

and also

△

⁢

A

⁢

O

⁢

E

≅

△

⁢

A

⁢

O

⁢

F

. Thus

F

⁢

O

=

E

⁢

O

=

D

⁢

O

. It follows that

O

is the incenter of

△

⁢

A

⁢

B

⁢

C

since its

distance

from all three sides is equal.

Also, since

F

⁢

O

=

D

⁢

O

we see that

△

⁢

B

⁢

O

⁢

F

and

△

⁢

B

⁢

O

⁢

D

are

right triangles

with two equal sides, so by SSA (which is applicable for right triangles),

△

⁢

B

⁢

O

⁢

F

≅

△

⁢

B

⁢

O

⁢

D

. Thus

B

⁢

O

bisects

∠

⁢

A

⁢

B

⁢

C

.

Title

proof of triangle incenter

Canonical name

ProofOfTriangleIncenter

Date of creation

2013-03-22 17:12:26

Last modified on

2013-03-22 17:12:26

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Proof

Classification

msc 51M99

Generated on Fri Feb 9 22:09:39 2018 by

LaTeXML
SSA

SSA

SSA

is a method for determining whether two

triangles

are

congruent

by comparing two sides and a non-inclusive angle. However, unlike SAS, SSS, ASA, and SAA, this does not prove

congruence

in all cases.

Suppose we have two triangles,

△

⁢

A

⁢

B

⁢

C

and

△

⁢

P

⁢

Q

⁢

R

.

△

⁢

A

⁢

B

⁢

C

≅

?

⁢

△

⁢

P

⁢

Q

⁢

R

if

A

⁢

B

¯

≅

P

⁢

Q

¯

,

B

⁢

C

¯

≅

Q

⁢

R

¯

, and either

∠

⁢

B

⁢

A

⁢

C

≅

∠

⁢

Q

⁢

P

⁢

R

or

∠

⁢

B

⁢

C

⁢

A

≅

∠

⁢

Q

⁢

R

⁢

P

.

Since this method does not prove congruence, it is more useful for disproving it. If the SSA method is attempted between

△

⁢

A

⁢

B

⁢

C

and

△

⁢

P

⁢

Q

⁢

R

and fails for every

A

⁢

B

⁢

C

,

B

⁢

C

⁢

A

, and

C

⁢

B

⁢

A

against every

P

⁢

Q

⁢

R

,

Q

⁢

R

⁢

P

, and

R

⁢

P

⁢

Q

, then

△

⁢

A

⁢

B

⁢

C

≇

△

⁢

P

⁢

Q

⁢

R

.

Suppose

△

⁢

A

⁢

B

⁢

C

and

△

⁢

P

⁢

Q

⁢

R

the SSA test. The specific case where SSA fails, known as the ambiguous case, occurs if the congruent angles,

∠

⁢

B

⁢

A

⁢

C

and

∠

⁢

Q

⁢

P

⁢

R

, are acute. Let us illustrate this.

Suppose we have a

right triangle

,

△

⁢

X

⁢

Y

⁢

Z

, with

right angle

∠

⁢

X

⁢

Z

⁢

Y

. Let

P

and

Q

be two points on

X

⁢

Z

↔

equidistant from

Z

such that

P

is between

X

and

Z

and

Q

is not. Since

∠

⁢

X

⁢

Z

⁢

Y

is right, this makes

∠

⁢

P

⁢

Z

⁢

Y

right, and

P

,

Q

are equidistant from

Z

, thus

Y

⁢

Z

↔

bisects

P

and

Q

, and as such, every point on that line is equidistant from

P

and

Q

. From this, we know

Y

is equidistant from

P

and

Q

, thus

Y

⁢

P

¯

≅

Y

⁢

Q

¯

. Further,

∠

⁢

Y

⁢

X

⁢

P

is in fact the same angle as

∠

⁢

Y

⁢

X

⁢

Q

, thus

∠

⁢

Y

⁢

X

⁢

P

≅

∠

⁢

Y

⁢

X

⁢

Q

. Since

X

⁢

Y

¯

≅

X

⁢

Y

¯

,

△

⁢

X

⁢

Y

⁢

P

and

△

⁢

X

⁢

Y

⁢

Q

clearly meet the SSA test, and yet, just as clearly, are not congruent. This results from

∠

⁢

Y

⁢

X

⁢

Z

being acute. This example also reveals the exception to the ambiguous case, namely

△

⁢

X

⁢

Y

⁢

Z

. If

R

is a point on

X

⁢

Z

↔

such that

Y

⁢

R

¯

≅

Y

⁢

Z

¯

, then

R

≅

Z

. Proving this exception amounts to determining that

∠

⁢

X

⁢

Z

⁢

Y

is right, in which case the congruency could be proven instead with SAA.

However, if the congruent angles are not acute, i.e., they are either right or obtuse, then SSA is definitive.

Title

SSA

Canonical name

SSA

Date of creation

2013-03-22 12:28:53

Last modified on

2013-03-22 12:28:53

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 51M99

Generated on Fri Feb 9 22:09:41 2018 by

LaTeXML
secant line

secant line

The

secant line

(or simply the

secant

) of a curve is a

straight line

intersecting the curve in at least two distinct points.  [The name is initially a participial form of the Latin verb

secare

‘’.]

If one sets a secant e.g. to the “

cubic parabola

”

y

=

x

3

through its points

(

0

,

0

)

and

(

1

,

1

)

,  there is also a third common point

(

-

1

,

-

1

)

.

Notice that a secant line can also be

tangent

to the curve at some point, given that tangency is only a local property.  In the following picture,

l

is a secant line for the curve

C

(since it intersects the curve at points

A

and

B

), yet it is also a

tangent line

at the point

A

.

Title

secant line

Canonical name

SecantLine

Date of creation

2013-03-22 14:50:34

Last modified on

2013-03-22 14:50:34

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

15

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 51M99

Synonym

secant

Synonym

secant of the curve

Synonym

secant to the curve

Related topic

curve

Defines

cubic parabola

Generated on Fri Feb 9 22:09:43 2018 by

LaTeXML
symmedian

symmedian

On any

triangle

, the three lines obtained by reflecting the medians in the (internal)

angle bisectors

are called the

symmedians

of the triangle.

In the picture,

B

⁢

X

is angle bisector and

B

⁢

M

a median. The

reflection

of

B

⁢

M

on

B

⁢

X

is

B

⁢

N

, a symmedian.

It can be stated as symmedians are

isogonal conjugates

of medians.

Title

symmedian

Canonical

name

Symmedian

Date of creation

2013-03-22 12:10:55

Last modified on

2013-03-22 12:10:55

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 51M99

Related topic

Triangle

Related topic

LemoinePoint

Related topic

GergonnePoint

Related topic

Isogonal

Related topic

IsogonalConjugate

Related topic

FundamentalTheoremOnIsogonalLines

Related topic

LemoineCircle

Generated on Fri Feb 9 22:09:45 2018 by

LaTeXML
arclength as filtered limit

arclength as filtered limit

The length (

http://planetmath.org/

Rectifiable

) of a rectifiable curve may be phrased as a filtered limit.
To do this, we will define a filter of

partitions

of an

interval

[

a

,

b

]

. Let

𝐏

be the set of all ordered tuplets of distinct
elements of

[

a

,

b

]

whose entries are

increasing

:

𝐏

=

{

(

t

1

,

…

t

n

)

∣

(

a

≤

t

1

<

t

2

<

⋯

<

t

n

≤

b

)

∧

(

n

∈

ℤ

)

∧

(

n

>

0

)

}

We shall refer to elements of

𝐏

as partitions of the interval

[

a

,

b

]

. We shall say that

(

t

1

,

…

,

t

n

)

is a

refinement

of a
partition

(

s

1

,

…

,

s

m

)

if

{

t

1

,

…

,

t

n

}

⊃

{

s

1

,

…

,

s

m

}

. Let

𝐅

⊂

𝒫

⁢

(

𝐏

)

be the
set of all subsets of

𝐏

such that, if a certain partition
belongs to

𝐅

then so do all refinements of that partition.

Let us see that

𝐅

is a

filter basis

. Suppose that

A

and

B

are elements of

𝐅

. If a partition belongs to both

A

and

B

then every one of its refinements will also belong to both

A

and

B

, hence

A

∩

B

∈

𝐅

.

Next, note that, if a partition of

B

is a refinement of a partition
of

A

then, by the

triangle inequality

, the length of

Π

⁢

(

B

)

is

greater

than the length of

Π

⁢

(

A

)

. By

definition

, for every

ϵ

>

0

, we can pick a partition

A

such that the length of

Π

⁢

(

A

)

differs from the length of the curve by at most

ϵ

.
Since the length of

Π

⁢

(

B

)

for any partition

B

refining

A

lies
between the length of

Π

⁢

(

A

)

and the length of the curve, we see
that the length of

Π

⁢

(

B

)

will also differ by at most

ϵ

, so
the length of the curve is the limit of the length of polygonal lines
according to the filter generated by

𝐅

.

Title

arclength as filtered limit

Canonical

name

ArclengthAsFilteredLimit

Date of creation

2013-03-22 15:49:34

Last modified on

2013-03-22 15:49:34

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Result

Classification

msc 51N05

Generated on Fri Feb 9 22:09:46 2018 by

LaTeXML
diagonal

diagonal

Let

P

be a

polygon

or a

polyhedron

. Two vertices on

P

are

adjacent

if the

line segment

joining them is an edge of

P

. A

diagonal

of

P

is a line segment joining two non-adjacent vertices.

Below is a figure showing a

hexagon

and all its diagonals (in red) with

X

as one of its

endpoints

.

Remarks

.

•

If

P

is convex, then the

relative interior

of a diagonal lies in the relative interior of

P

. Below is a figure showing that a diagonal may partially lie outside of

P

.

\begin{pspicture}(-227.62204pt,0.0pt)(0.0pt,56.905502pt)\leavevmode%
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces%
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces%
\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces\ignorespaces%
\special{pst:
\pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end
}\ignorespaces\end{pspicture}

•

If a polygon

P

has

n

(distinct) vertices, then it has

n

⁢

(

n

-

3

)

2

diagonals.

Title

diagonal

Canonical

name

Diagonal

Date of creation

2013-03-22 17:34:41

Last modified on

2013-03-22 17:34:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51N05

Related topic

BasicPolygon

Related topic

Polyhedron

Defines

adjacent vertices

Generated on Fri Feb 9 22:10:06 2018 by

LaTeXML
locus

locus

Locus

(from Latin

locus

‘’) is a set of points satisfying a certain condition and such that no point outside the set

satisfies

this condition.

For example, in the plane the locus of such point that is at a fixed

distance

r

from a

fixed point

O

is the circle with centre

O

and radius

r

; the locus of such point which is equidistant from both ends of a

line segment

is the

perpendicular bisector

of this line segment.

In general, the loci in the plane are geometric curves; thus one may speak of

locus curves

.

Title

locus

Canonical name

Locus

Date of creation

2013-03-22 17:07:52

Last modified on

2013-03-22 17:07:52

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N05

Related topic

SetMembership

Defines

locus

Defines

loci

Generated on Fri Feb 9 22:10:12 2018 by

LaTeXML
NURBS curve

NURBS curve

1

Introduction

A

NURBS curve

, which is an acronym for

Non-Uniform Rational B-Spline curve

, is a

generalization

of both Bézier (

http://planetmath.org/BezierCurve

) and BSpline (

http://planetmath.org/B-splines

) curves. NURBS are commonly used in computer graphics, computer-aided design (CAD), engineering (CAE), and manufacturing (CAM).

2

Definition

A NURBS curve is a parametric curve defined by its , a set of weighted control points, and a knot vector. It is defined as

c

⁢

(

u

)

=

∑

i

=

0

n

N

i

,

p

⁢

(

u

)

⁢

w

i

⁢

P

i

∑

i

=

0

n

N

i

,

p

⁢

(

u

)

⁢

w

i

0

≤

u

≤

1

where

u

is the

parameter

,

p

is the ,

N

i

,

p

are the B-spline basis functions,

P

i

are the control points and

w

i

are the weights.

Title

NURBS curve

Canonical

name

NURBSCurve

Date of creation

2013-03-22 17:10:59

Last modified on

2013-03-22 17:10:59

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

12

Author

stitch (17269)

Entry type

Definition

Classification

msc 51N05

Synonym

nonuniform rational B-spline curve

Related topic

BezierCurve

Related topic

BSpline

Related topic

NURBSSurface

Generated on Fri Feb 9 22:10:14 2018 by

LaTeXML