aarjaneiro/planetmath huggingface dataset:
NURBS surface

NURBS surface

1

Introduction

A

NURBS surface

, which is an acronym for

Non-Uniform Rational B-Spline surface

, is a

generalization

of both Bézier (

http://planetmath.org/BezierCurve

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

2

Definition

A NURBS surface is parametric surface defined by its , an array of

n

+

1

rows and

m

+

1

columns weighted control points and a knot vector in each direction. It is defined as

c

⁢

(

u

,

v

)

=

∑

i

=

0

n

∑

j

=

0

m

N

i

,

p

⁢

(

u

)

⁢

N

j

,

q

⁢

(

v

)

⁢

w

i

,

j

⁢

P

i

,

j

∑

i

=

0

n

∑

j

=

0

m

N

i

,

p

⁢

(

u

)

⁢

N

j

,

q

⁢

(

v

)

⁢

w

i

,

j

0

≤

u

≤

1

,

0

≤

v

≤

1

where

u

and

v

are the

parameters

in each direction,

p

is the in the

u

-direction,

q

is the in the

v

-direction,

N

i

,

p

and

N

j

,

q

are the B-spline basis functions,

P

i

,

j

are the control points and

w

i

,

j

are the weights.

Title

NURBS surface

Canonical

name

NURBSSurface

Date of creation

2013-03-22 17:23:51

Last modified on

2013-03-22 17:23:51

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

6

Author

stitch (17269)

Entry type

Definition

Classification

msc 51N05

Synonym

nonuniform rational B-spline surface

Related topic

NURBS

Generated on Fri Feb 9 22:10:16 2018 by

LaTeXML
osculating curve

osculating curve

Definition.

From a family of

plane curves

, the

osculating curve

of the curve

y

=

f

⁢

(

x

)

in a certain point is the curve of the family which has the highest

order contact

with the curve

y

=

f

⁢

(

x

)

in that point.

Example 1.

From the family of the graphs of the polynomial functions

P

n

⁢

(

x

)

:=

c

0

+

c

1

⁢

(

x

-

x

0

)

+

…

+

c

n

⁢

(

x

-

x

0

)

n

,

the osculating curve of

y

=

f

⁢

(

x

)

in

(

x

0

,

f

⁢

(

x

0

)

)

is the

Taylor polynomial

of degree

n

of the

function

f

.

Example 2.

Determine the osculating

hyperbola

with axes

parallel

to the

coordinate

axes for the curve

y

=

cos

⁡

x

in the point

(

0

,

1

)

.  What is the order of contact?

We may seek the osculating hyperbola from the three-parametric family

x

2

a

2

-

(

y

-

y

0

)

2

b

2

=

-

1

.

(1)

Removing the denominators and differentiating six times successively yield the equations

{

b

2

⁢

x

2

-

a

2

⁢

(

y

-

y

0

)

2

+

a

2

⁢

b

2

=

0

,

b

2

⁢

x

-

a

2

⁢

(

y

-

y

0

)

⁢

y

′

=

0

,

b

2

-

a

2

⁢

y

′

⁣

2

-

a

2

⁢

(

y

-

y

0

)

⁢

y

′′

=

0

,

3

⁢

y

′

⁢

y

′′

+

(

y

-

y

0

)

⁢

y

′′′

=

0

,

3

⁢

y

′′

⁣

2

+

4

⁢

y

′

⁢

y

′′′

+

(

y

-

y

0

)

⁢

y

′′′′

=

0

,

10

⁢

y

′′

⁢

y

′′′

+

5

⁢

y

′

⁢

y

′′′′

+

(

y

-

y

0

)

⁢

y

(

5

)

=

0

,

10

⁢

y

′′′

⁣

2

+

15

⁢

y

′′

⁢

y

′′′′

+

6

⁢

y

′

⁢

y

(

5

)

+

(

y

-

y

0

)

⁢

y

(

6

)

=

0

.

(2)

Into these equations we can substitute the coordinates

x

=

0

,

y

=

1

of the contact point and the values of the

derivatives

y

′

=

-

sin

⁡

x

,

y

′′

=

-

cos

⁡

x

,

y

′′′

=

sin

⁡

x

,

y

′′′′

=

cos

⁡

x

,

y

(

5

)

=

-

sin

⁡

x

,

y

(

6

)

=

-

cos

⁡

x

of cosine in that point; the values are

0

,

-

1

,

0

,

1

,

0

,

-

1

.  The first, third and fifth of the equations (2) give the result

y

0

=

4

,

b

2

=

9

,

a

2

=

3

, whence the osculating hyperbola is

x

2

3

-

(

y

-

4

)

2

9

=

-

1

.

When we substitute the pertinent values of the cosine derivatives into the two last equations (2), we see that only the former of them is satisfied.  It means that the order of contact between the cosine curve and the hyperbola is 5.

Example 3.

The osculating

parabola

of the

exponential

curve (

http://planetmath.org/ExponentialFunction

)

y

=

e

x

in the point

(

0

,

1

)

is

4

⁢

x

2

+

y

2

+

4

⁢

x

⁢

y

+

14

⁢

x

-

20

⁢

y

+

19

=

0

.

The order of contact is only 3.

Title

osculating curve

Canonical name

OsculatingCurve

Date of creation

2013-03-22 17:57:17

Last modified on

2013-03-22 17:57:17

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N05

Classification

msc 53A04

Related topic

OrderOfVanishing

Related topic

CircleOfCurvature

Related topic

Cosine

Related topic

QuadraticCurves

Generated on Fri Feb 9 22:10:19 2018 by

LaTeXML
parallel curve

parallel curve

Given two curves, one is a

parallel curve

(also known as an

offset curve

) of the other if the points on the first curve are equidistant to the corresponding points in the direction of the second curve’s normal. Alternatively, a

parallel

of a curve can be defined as the

envelope

of

congruent

circles whose centers lie on the curve.

For a parametric curve in the plane defined by

F

→

⁢

(

u

)

:=

(

x

⁢

(

u

)

,

y

⁢

(

u

)

)

,  its parallel curve

G

→

⁢

(

u

)

:=

(

X

⁢

(

u

)

,

Y

⁢

(

u

)

)

with offset

t

is defined by

X

⁢

(

u

)

=

x

⁢

(

u

)

+

t

⁢

y

′

⁢

(

u

)

x

′

⁢

(

u

)

2

+

y

′

⁢

(

u

)

2

Y

⁢

(

u

)

=

y

⁢

(

u

)

-

t

⁢

x

′

⁢

(

u

)

x

′

⁢

(

u

)

2

+

y

′

⁢

(

u

)

2

0.1

Examples

The most

elementary

example of parallel curves is given by the family of

concentric circles

X

⁢

(

u

)

=

t

⁢

cos

⁡

u

Y

⁢

(

u

)

=

t

⁢

sin

⁡

u

Except for

trivial

cases such as circles and lines, parallel curves may be quite different from the original curve as the offset gets larger. An example of this is given by the

catenary

x

⁢

(

u

)

=

u

y

⁢

(

u

)

=

cosh

⁡

u

From the definition, the family of parallel curves is then

X

=

u

+

t

⁢

sinh

⁡

u

1

+

sinh

2

⁡

u

=

u

+

t

⁢

tanh

⁡

u

Y

=

cosh

⁡

u

-

t

1

+

sinh

2

⁡

u

=

cosh

⁡

u

-

t

cosh

⁡

u

where

t

=

0

correspond to the catenary.

Eliminating the parameter

u

from these

equations

; the latter gives

cosh

⁡

u

=

Y

+

Y

2

+

4

⁢

t

2

, i.e.

u

=

arcosh

⁡

Y

+

Y

2

+

4

⁢

t

2

. Thus we obtain the implicit

representation

arcosh

⁡

Y

+

Y

2

+

4

⁢

t

2

+

t

⁢

tanh

⁡

(

arcosh

⁡

Y

+

Y

2

+

4

⁢

t

2

)

-

X

=

0

Title

parallel curve

Canonical name

ParallelCurve

Date of creation

2013-03-22 17:13:10

Last modified on

2013-03-22 17:13:10

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

21

Author

stitch (17269)

Entry type

Definition

Classification

msc 51N05

Synonym

offset curve

Related topic

ParallellismInEuclideanPlane

Related topic

NormalLine

Related topic

HyperbolicFunctions

Generated on Fri Feb 9 22:10:23 2018 by

LaTeXML
piecewise smooth

piecewise smooth

A curve

α

:

[

a

,

b

]

→

ℝ

n

is said to be

piecewise smooth

if each

component

α

1

,

…

,

α

n

of

α

has a

bounded

derivative

α

i

′

(

i

=

1

,

…

,

n

)

which is

continuous

everywhere in

[

a

,

b

]

except (possibly) at a finite number of points at which left- and

right-sided derivatives

exist.

•

Every piecewise smooth curve is rectifiable.

•

Every rectifiable curve can be approximated by piecewise smooth curves.

Title

piecewise smooth

Canonical name

PiecewiseSmooth

Date of creation

2013-03-22 12:53:26

Last modified on

2013-03-22 12:53:26

Owner

cvalente (11260)

Last modified by

cvalente (11260)

Numerical id

7

Author

cvalente (11260)

Entry type

Definition

Classification

msc 51N05

Related topic

Rectifiable

Generated on Fri Feb 9 22:10:24 2018 by

LaTeXML
properties of parallel curves

properties of parallel curves

•

Two

plane curves

are

parallel curves

of each other, if every normal of one curve is also a normal of the other curve (then one may show that the

distance

of the corresponding points of the curves is a ).

•

Two curves are parallel curves of each other, if they are the loci of the

end points

of a

line segment

which moves perpendicularly to its own direction.

•

Every regular curve having a

continuous

curvature

has an

infinite

family of parallel curves.

•

The

parallelism

of curves is an

equivalence relation

.

•

The two parallel curves

γ

±

a

on both sides of a curve

γ

at the distance

a

form the

envelope

of the family of circles with center on

γ

and radius

a

.

•

If

γ

is a and closed curve with

perimeter

p

, then the perimeter of

γ

±

a

is equal to

p

±

2

⁢

π

⁢

a

and the area between

γ

and the parallel curve is equal to

p

⁢

a

±

π

⁢

a

2

(one must also assume that the parallel curve don’t

intersect

the evolute of

γ

).

Title

properties of parallel curves

Canonical name

PropertiesOfParallelCurves

Date of creation

2013-03-22 17:14:30

Last modified on

2013-03-22 17:14:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Topic

Classification

msc 51N05

Generated on Fri Feb 9 22:10:26 2018 by

LaTeXML
rectifiable curve

rectifiable curve

Definitions

Let

f

:

[

a

,

b

]

→

ℝ

k

be a

simple curve

in

ℝ

k

and let

P

=

(

s

0

,

…

,

s

n

)

with

a

≤

s

0

<

s

1

<

⋯

<

s

n

≤

b

be a

partition

of the

interval

[

a

,

b

]

; then the points in the set

{

f

⁢

(

s

0

)

,

f

⁢

(

s

1

)

,

…

,

f

⁢

(

s

n

)

}

are called the

vertices of the

inscribed

polygonal path

Π

⁢

(

P

)

determined by

P

.
The

length

of the inscribed polygonal path
is defined to be

∑

m

=

1

n

|

f

⁢

(

s

m

)

-

f

⁢

(

s

m

-

1

)

|

.
The simple curve is said to be

rectifiable

if there exists a positive number

M

such that the length of the inscribed polygonal path

Π

⁢

(

P

)

is less than

M

for all possible partitions

P

of

[

a

,

b

]

.
If the simple curve is rectifiable then its

length

is defined as the least

upper bound

of the lengths of inscribed polygonal paths taken over all possible partitions.

Notes

The intuition underlying this

definition

is that, to measure the length of a curve, one could approximate the curve by an inscribed polygonal path. As one increases the number of vertices of the polygonal path, one expects to obtain better approximations to the curve, and hence one would expect the length of the curve to be the limit of the length of these polygonal paths. Since the length of a polygonal path increases as one adds vertices, this means we are dealing with the limit of an increasing quantity, which will equal its supremum.

If one does not demand the curve to be simple (that is, that

f

be injective), then one may still proceed as above, taking the supremum of the lengths of all possible inscribed polygonal paths. However, what one will obtain is not necessarily the length of the curve, but the total

distance

travelled along the curve. For instance, if a portion of the curve is traced more than once, then the length of that portion of the curve will be counted more than once.

Although this definition mentions a parameterization of the (simple) curve, what is being defined actually is

independent

of the choice of parameterization. In fact, all the information that is used is the

ordering

of points along the curve, which is invariant under reparameterization. Also, because the length of a

line segment

does not depend on how one might choose to orient the line segment, what is being defined here is invariant under reversing the parameterization of the curve. Hence, this is a geometrical

property

of the curve.

Title

rectifiable curve

Canonical

name

RectifiableCurve

Date of creation

2013-03-22 12:53:23

Last modified on

2013-03-22 12:53:23

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Definition

Classification

msc 51N05

Related topic

ArcLength

Related topic

PiecewiseSmooth

Related topic

StraightLineIsShortestCurveBetweenTwoPoints

Related topic

IntegralRepresentationOfLengthOfSmoothCurve

Defines

rectifiable

Defines

length

Generated on Fri Feb 9 22:10:28 2018 by

LaTeXML
straight line is shortest curve between two points

straight line is shortest curve between two points

Suppose

p

and

q

are two distinct points in

ℝ

n

,
and

γ

is a

rectifiable curve

from

p

to

q

.
Then every curve other than the

straight line

segment

from

p

to

q

has a length

greater

than the

Euclidean distance

∥

p

-

q

∥

.

Proof.

Let

γ

:

[

0

,

1

]

→

ℝ

n

be the curve with length

L

.
If it is not

straight

1

1

If

γ

is a straight line segment but is not

injective

, that is, it moves

p

and

q

, then it is obvious that

L

>

∥

p

-

q

∥

.

, then there exists a point

x

=

γ

⁢

(

t

)

that does not lie on the

line segment

from

p

to

q

.
We have

L

≥

∥

q

-

x

∥

+

∥

x

-

p

∥

>

∥

p

-

q

∥

.

The first

inequality

comes from the definition of

L

as the least

upper bound

of the length of

any

broken-line approximation to the curve

γ

.
The second inequality is the usual

triangle inequality

,
but it is a

strict inequality

since

x

lies outside the line segment between

p

and

q

,
as shown in the following diagram.
∎

Title

straight line is shortest curve between two points

Canonical name

StraightLineIsShortestCurveBetweenTwoPoints

Date of creation

2013-03-22 15:39:43

Last modified on

2013-03-22 15:39:43

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

11

Author

stevecheng (10074)

Entry type

Result

Classification

msc 51N05

Related topic

ArcLength

Related topic

Rectifiable

Generated on Fri Feb 9 22:10:30 2018 by

LaTeXML
meromorphic function on projective space must be rational

meromorphic function on projective space must be rational

To define a rational function on

complex projective space

ℙ

n

, we just take
two

homogeneous polynomials

of the same degree

p

and

q

on

ℂ

n

+

1

,

and we
note that

p

/

q

induces a

meromorphic function

on

ℙ

n

.

In fact, every
meromorphic function on

ℙ

n

is

rational

.

Theorem

.

Let

f

be a meromorphic function on

P

n

.
Then

f

is rational.

Proof.

Note that the

zero set

of

f

and the pole set are

analytic

subvarieties of

ℙ

n

and hence

algebraic

by Chow’s theorem.

f

induces a meromorphic function

f

~

on

ℂ

n

+

1

∖

{

0

}

. Let

p

and

q

be two homogeneous polynomials
such that

q

=

0

are the poles and

p

=

0

are the zeros of

f

~

.

We can assume we can take

p

and

q

such that if we multiply

f

~

by

q

/

p

we have a

holomorphic function

outside the origin. Hence

(

q

/

p

)

⁢

f

~

extends through the origin
by Hartogs’ theorem. Further since

f

~

was
constant on

complex lines

through the origin, it is not hard to see that

(

q

/

p

)

⁢

f

~

is

homogeneous

and hence a homogeneous polynomial, by the same

argument

as in the proof of Chow’s theorem.
Since it is not zero outside the origin, it can’t be zero at the origin, and hence

(

q

/

p

)

⁢

f

~

must be a constant, and the proof is finished.
∎

Title

meromorphic function on projective space must be rational

Canonical name

MeromorphicFunctionOnProjectiveSpaceMustBeRational

Date of creation

2013-03-22 17:52:13

Last modified on

2013-03-22 17:52:13

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Theorem

Classification

msc 51N15

Classification

msc 32A20

Related topic

ChowsTheorem

Generated on Fri Feb 9 22:10:44 2018 by

LaTeXML
analytic geometry

analytic geometry

Analytic geometry

is the of

geometry

that uses

mathematical analysis

and calculations for investigating geometric problems. Many such problems can be put into the form of equations, and, by analyzing these equations, one may obtain solutions which can be interpreted geometrically.

The fundamental idea behind analytic geometry is that the position of any point on a plane can be given by an ordered pair of

real numbers

and any point in space by an ordered triple of real numbers; for this purpose, one has to have a

coordinate system

which determines the values of the numbers which serve as the

coordinates

of the points.
One of the coordinate system often used in mathematics and physics is the

Cartesian

system which employs three

orthogonal

axes. The correspondence of the points on a plane and the ordered pairs (and similarly the points in space and the ordered triples) is a

bijection

. A locus condition for a line or a curve on the plane as well as for a line, a curve, or a surface in space may then be expressed as an equation or a system of equations.

For example, if we want to study the line which

passes through

the points corresponding to the
ordered pairs (

http://

planetmath

.org/CartesianProduct

)

(

5

,

0

)

and

(

0

,

8

)

,  one can infer that all points

(

x

,

y

)

of this line satisfy the equation

x

5

+

y

8

=

1

.

The

Cartesian coordinate system

was introduced and applied by the French mathematician and philosopher René Descartes in 1637 in his work

Géométrie

. Thus, Descartes is considered to be the founder of analytic geometry.

Title

analytic geometry

Canonical name

AnalyticGeometry

Date of creation

2013-03-22 17:36:23

Last modified on

2013-03-22 17:36:23

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Topic

Classification

msc 51N20

Classification

msc 01A45

Related topic

CartesianCoordinates

Related topic

EuclideanGeometryOfThePlane

Related topic

EuclideanGeometryOfTheSpace

Related topic

LineInThePlane

Related topic

LineInSpace

Related topic

EquationOfPlane

Related topic

Ellipse

Related topic

Hyperbola

Related topic

Parabola

Related topic

QuadraticSurfaces

Related topic

CissoidOfDiocles

Related topic

GeneratricesOfOneSheetedHyperboloid

Related topic

ConicSection

Generated on Fri Feb 9 22:10:46 2018 by

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angle between line and plane

angle between line and plane

The

angle between a line

l

and a plane

τ

is defined as the

least possible

angle

ω

between

l

and a line contained by

τ

.

It is apparent that

ω

satisfies always

0

≦

ω

≦

90

∘

.

Let the plane

τ

be given by the equation (

http://planetmath.org/EquationOfPlane

)

A

⁢

x

+

B

⁢

y

+

C

⁢

z

+

D

=

0

,  i.e. its

normal vector

has the

components

A

,

B

,

C

. Let a

direction vector

of the line

l

have the components

a

,

b

,

c

. Then the angle

ω

between

l

and

τ

is obtained from the equation

sin

⁡

ω

=

|

A

⁢

a

+

B

⁢

b

+

C

⁢

c

|

A

2

+

B

2

+

C

2

⁢

a

2

+

b

2

+

c

2

.

In fact, the right hand side (

http://planetmath.org/Equation

) is the cosine of the angle

α

between

l

and the

surface normal

of

τ

(see

angle between two lines

), and

ω

is the

complementary angle

of

α

.

Example.

Consider the

x

⁢

y

-plane and the line

l

through the origin and the point

(

1

,

1

,

1

)

. We can use the components

1

,

1

,

1

for the direction vector of

l

and the components

0

,

0

,

1

for the normal vector of the plane. We have

ω

=

arcsin

⁡

1

⋅

0

+

1

⋅

0

+

1

⋅

1

1

2

+

1

2

+

1

2

⁢

0

2

+

0

2

+

1

2

=

arcsin

⁡

1

3

≈

35.26

∘

.

Title

angle between line and plane

Canonical name

AngleBetweenLineAndPlane

Date of creation

2013-03-22 17:30:14

Last modified on

2013-03-22 17:30:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N20

Synonym

slant

Synonym

inclination

Related topic

AngleBetweenTwoLines

Related topic

DotProduct

Related topic

EquationOfPlane

Related topic

AngleBetweenTwoPlanes

Related topic

NormalOfPlane

Related topic

ProjectionOfRightAngle

Defines

angle between plane and line

Generated on Fri Feb 9 22:10:48 2018 by

LaTeXML
angle between two planes

angle between two planes

Let

π

1

and

π

2

be two planes in the three-dimensional

Euclidean space

ℝ

3

.  The angle

θ

between these planes is defined by means of the

normal vectors

𝒏

1

and

𝒏

2

of

π

1

and

π

2

through the relationship

cos

⁡

θ

=

|

⟨

𝒏

1

,

𝒏

2

⟩

∥

𝒏

1

∥

⁢

∥

𝒏

2

∥

|

,

where the

numerator

is the

inner product

of

𝒏

1

and

𝒏

2

and the denominator is product of the lengths of

𝒏

1

and

𝒏

2

.  The formula implies that the angle

θ

satisfies

0

≤

θ

≤

π

2

.

The quotient in the formula remains unchanged as one multiplies the normal vectors by some non-zero

real numbers

, so that the cosine is

independent

of the lengths of the chosen vectors.  Therefore, there is no ambiguity in this definition.

Figure 1:

Angle between two planes

Generalization

.  The above definition can be generalized, at least locally, to a pair of intersecting

differentiable

surfaces in

ℝ

3

.  Given two differentiable surfaces

S

1

and

S

2

and a point

p

∈

S

1

∩

S

2

, the angle between

S

1

and

S

2

at

p

is defined to be the angle between the

tangent planes

T

p

⁢

(

S

1

)

and

T

p

⁢

(

S

2

)

.

Title

angle between two planes

Canonical name

AngleBetweenTwoPlanes

Date of creation

2013-03-22 16:19:41

Last modified on

2013-03-22 16:19:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 51N20

Synonym

angle between planes

Related topic

AngleBetweenTwoLines

Related topic

AngleBetweenLineAndPlane

Defines

angle between differentiable surfaces

Generated on Fri Feb 9 22:10:59 2018 by

LaTeXML
angle bisector as locus

angle bisector as locus

If

0

<

α

<

180

o

,  then the

angle bisector

of

α

is the locus of all such points which are equidistant from both sides of the angle (it is proved by using the AAS and SSA

theorems

).

The equation of the angle bisectors of all four angles formed by two

intersecting lines

a

1

⁢

x

+

b

1

⁢

y

+

c

1

=

0

,

a

2

⁢

x

+

b

2

⁢

y

+

c

2

=

0

(1)

is

a

1

⁢

x

+

b

1

⁢

y

+

c

1

a

1

2

+

b

1

2

=

±

a

2

⁢

x

+

b

2

⁢

y

+

c

2

a

2

2

+

b

2

2

,

(2)

which may be written in the form

x

⁢

sin

⁡

α

1

-

y

⁢

cos

⁡

α

1

+

h

1

=

±

(

x

⁢

sin

⁡

α

2

-

y

⁢

cos

⁡

α

2

+

h

2

)

(3)

after performing the

divisions

in (2) termwise; the angles

α

1

and

α

2

then the

slope angles

of the lines.

Note.

The two lines in (2) are

perpendicular

, since their slopes

sin

⁡

α

1

±

sin

⁡

α

2

cos

⁡

α

1

±

cos

⁡

α

2

are opposite

inverses

of each other.

Title

angle bisector as locus

Canonical

name

AngleBisectorAsLocus

Date of creation

2013-03-22 17:10:41

Last modified on

2013-03-22 17:10:41

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Definition

Classification

msc 51N20

Related topic

DistanceFromPointToALine

Related topic

ConverseOfIsoscelesTriangleTheorem

Related topic

ConstructionOfTangent

Related topic

LengthsOfAngleBisectors

Related topic

Incenter

Related topic

CenterNormalAndCenterNormalPlaneAsLoci

Generated on Fri Feb 9 22:11:01 2018 by

LaTeXML