orthogonal circles

orthogonal circles

Two circles intersecting orthogonally (

http://planetmath.org/ConvexAngle

) are orthogonal curves and called

orthogonal circles

of each other.

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Since the

tangent of circle

is

perpendicular

to the radius drawn to the tangency point, the both radii of two orthogonal circles drawn to the point of

intersection

and the

line segment

connecting the centres form a

right triangle

. If

(

x

-

a

1

)

2

+

(

y

-

b

1

)

2

=

r

1

2

and

(

x

-

a

2

)

2

+

(

y

-

b

2

)

2

=

r

2

2

are the equations of the circles, then, by

Pythagorean theorem

,

r

1

2

+

r

2

2

=

(

a

2

-

a

1

)

2

+

(

b

2

-

b

1

)

2

(1)

is the condition of the orthogonality (

http://planetmath.org/OrthogonalCurves

) of those circles.

The equation (1) tells that the centre of one circle is always outside its orthogonal circle.  If

(

x

0

,

y

0

)

is an arbitrary point outside the circle

(

x

-

a

)

2

+

(

y

-

b

)

2

=

r

2

,  one can always draw with that point as centre the orthogonal circle of this circle:  its radius is the

limited tangent

from

(

x

0

,

y

0

)

to the given circle. The square (

http://planetmath.org/SquareOfANumber

) of the limited tangent is equal to the power of the point with respect to the circle and thus

(

x

0

-

a

)

2

+

(

y

0

-

b

)

2

-

r

2

.  Accordingly, the equation of the orthogonal circle is

(

x

-

x

0

)

2

+

(

y

-

y

0

)

2

=

(

x

0

-

a

)

2

+

(

y

0

-

b

)

2

-

r

2

.

One of two

orthogonal

(

http://planetmath.org/Orthogonal

) circles

harmonically

any

diameter

of the other circle.

Title

orthogonal circles

Canonical

name

OrthogonalCircles

Date of creation

2013-03-22 17:41:32

Last modified on

2013-03-22 17:41:32

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Topic

Classification

msc 51N20

Related topic

PerpendicularityInEuclideanPlane

Defines

orthogonal circle

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