Interval

java.lang.Object
- xal.tools.math.Interval

All Implemented Interfaces:

java.io.Serializable

Direct Known Subclasses:

ClosedInterval, OpenInterval
```
public class Interval
extends java.lang.Object
implements java.io.Serializable
```
Represents an interval on the real line. Intervals are identified by their end points, which may or may not be included depending upon whether the interval is open or closed. Nonetheless, we consider the left end the minimum value and the right end point the maximum value and refer to them as such.

Issues concerning boundary points are delegated to the derived classes.

See Also:

Serialized Form

Constructor Summary

Constructors
Constructor and Description
`Interval()` Default constructor - creates a new instance of Interval with one point, the origin.
`Interval(double min, double max)` Initializing constructor - create an interval with specified endpoints
`Interval(Interval I)` Copy constructor - create a new open interval initialized to the argument.

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`boolean`	`containsAE(Interval I)`
`Interval`	`convexHull(Interval I)` Compute and return the smallest interval containing both this interval and the argument interval (i.e., the union of they are intersected).
`static Interval`	`createFromEndpoints(double dblMin, double dblMax)` Creates a new interval object according to the given endpoints.
`static Interval`	`createFromMidpoint(double dblMidpt, double dblLng)` Creates an interval object given the midpoint (centroid) of the interval and the length of the interval (which is the Lebesgue measure).
`boolean`	`equals(Interval I)` Are intervals equal
`double`	`getMax()` Get maximum value of interval.
`double`	`getMin()` Get minimum value of interval.
`Interval`	`intersection(Interval I)` Compute and return the largest interval contained in this interval and the argument interval (i.e., the intersection).
`boolean`	`intersects(Interval I)` Inspects for non-empty intersection with the given interval.
`boolean`	`isBoundary(double x)` Is point a boundary element?
`double`	`measure()` Compute the interval length.
`boolean`	`membership(double x)` Is point a member of this interval I ⊂ R.
`double`	`midpoint()` Compute the interval midpoint.
`void`	`print(java.io.PrintWriter os)` Print out contents on an output stream
`void`	`println(java.io.PrintWriter os)` Print out contents on an output stream, terminate in newline character
`void`	`setMax(double max)` Set the right end point
`void`	`setMin(double min)` Set the left end point
`java.lang.String`	`toString()` Return the contents of the interval as a `String`.
`double[]`	`vertexCoordinates(double x)` Compute the local "vertex coordinates" of the argument x with respect to this interval I ⊂ R.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail
- Interval
```
public Interval()
```
  Default constructor - creates a new instance of Interval with one point, the origin.
- Interval
```
public Interval(double min,
                double max)
         throws MathException
```
  Initializing constructor - create an interval with specified endpoints
  
  Parameters:
  
  min - left endpoint
  
  max - right endpoint
  
  Throws:
  
  MathException - max is smaller than min
- Interval
```
public Interval(Interval I)
         throws MathException
```
  Copy constructor - create a new open interval initialized to the argument.
  
  Parameters:
  
  I - interval to copy
  
  Throws:
  
  MathException - malformed interval to copy

Method Detail

createFromMidpoint
```
public static Interval createFromMidpoint(double dblMidpt,
                                          double dblLng)
```
Creates an interval object given the midpoint (centroid) of the interval and the length of the interval (which is the Lebesgue measure). We do not throw an exception assuming that both arguments are positive.

Parameters:

dblMidpt - the center location of the interval

dblLng - the length of the interval

Returns:

new interval object with endpoints producing the given center and length

Since:

Apr 28, 2011

createFromEndpoints
```
public static Interval createFromEndpoints(double dblMin,
                                           double dblMax)
```
Creates a new interval object according to the given endpoints. We do not throw an exception here, which would normally happen if the left-hand endpoint was larger than the right-hand endpoint. Instead we return a null value in this case.

Parameters:

dblMin - the left-hand endpoint

dblMax - the right-hand endpoint

Returns:

the interval [dblMin,dblMax]

Since:

Apr 28, 2011

setMin
```
public void setMin(double min)
```
Set the left end point

Parameters:

min - interval left-hand side

setMax
```
public void setMax(double max)
```
Set the right end point

Parameters:

max - interval right-hand side

getMin
```
public double getMin()
```
Get minimum value of interval.

Returns:

the left end point

getMax
```
public double getMax()
```
Get maximum value of interval.

Returns:

the right end point

measure
```
public double measure()
```
Compute the interval length. This is the difference in endpoints.

Returns:

length of the interval

midpoint
```
public double midpoint()
```
Compute the interval midpoint. This is the average of the endpoints.

Returns:

interval center of mass

membership
```
public boolean membership(double x)
```
Is point a member of this interval I ⊂ R. The test is done assuming I is closed, so the endpoints are included.

Parameters:

x - point to be tested as a member of this interval

Returns:

true if x ∈ I, false otherwise

equals
```
public boolean equals(Interval I)
```
Are intervals equal

Parameters:

I - interval object to be checked for equality

Returns:

true if both objects are equal as intervals

intersects
```
public boolean intersects(Interval I)
```
Inspects for non-empty intersection with the given interval. The interval is assumed closed.

Parameters:

I - interval to inspect.

Returns:

true if this ∩ I ≠ {}, false otherwise

Since:

Jun 2, 2009

isBoundary
```
public boolean isBoundary(double x)
```
Is point a boundary element?

Parameters:

x - point to be tested as boundary element

Returns:

true if x is a boundary element

containsAE
```
public boolean containsAE(Interval I)
```
Contains Almost Everywhere - ⊂ a.e.

Checks whether or not the given interval is a proper subset of this interval, modulo the endpoints. Specifically, I must be contained within this interval modulo the endpoints. We ignore the endpoints and, thus, the condition of whether or not either interval is closed or open.

Parameters:

I - interval to be tested

Returns:

true if I ⊂ this, false otherwise

Since:

Apr 28, 2011

vertexCoordinates

public double[] vertexCoordinates(double x)
                           throws MathException

Compute the local "vertex coordinates" of the argument x with respect to this interval I ⊂ R. These "local coordinates" {λ₁,λ₂} of a point x ∈ I can computed by solving the following linear equation for the {λ_i} :

⌈	x₁	x₂	⌉
⌊	1	1	⌋

⌈	λ₁	⌉
\|	λ₂	\|

⌈	x	⌉
⌊	1	⌋

where p = x ∈ R.

NOTES:

· After solving the above equation for the {λ₁,λ₂} if there is a λ_i such that λ_i ∉ [0,1], then x is not in T; that is, p∉ T.

Parameters:: x - point in I
Returns:: order 2-array vertex coordinates {λ₁,λ₂}
Throws:: MathException - argument is not a member of this set

convexHull
```
public Interval convexHull(Interval I)
                    throws MathException
```
Compute and return the smallest interval containing both this interval and the argument interval (i.e., the union of they are intersected).

Parameters:

I - right-hand-side argument

Returns:

union of this and I

Throws:

MathException - empty intersection

intersection
```
public Interval intersection(Interval I)
```
Compute and return the largest interval contained in this interval and the argument interval (i.e., the intersection). If the intersection of the two intervals contains no points (i.e., it is the empty set), then a null value is returned. Since the empty is a valid result of intersection, this value should be considered in any robust implementation.

Parameters:

I - interval to be intersected with this interval

Returns:

intersection of this and I, or null if the empty set {}

toString
```
public java.lang.String toString()
```
Return the contents of the interval as a String. return string representation of interval

Overrides:

toString in class java.lang.Object

See Also:

Object.toString()

print
```
public void print(java.io.PrintWriter os)
```
Print out contents on an output stream

Parameters:

os - output stream receiving content dump

println
```
public void println(java.io.PrintWriter os)
```
Print out contents on an output stream, terminate in newline character

Parameters:

os - output stream receiving content dump

Class Interval

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

Interval

Interval

Interval

Method Detail

createFromMidpoint

createFromEndpoints

setMin

setMax

getMin

getMax

measure

midpoint

membership

equals

intersects

isBoundary

containsAE

Contains Almost Everywhere - ⊂ a.e.

vertexCoordinates

NOTES:

convexHull

intersection

toString

print

println