Gapotchenko.FX.Math.Intervals 2026.3.5

Overview

The module provides data structures and primitives for working with intervals. Interval is a mathematical structure that defines a range of values in a unified and formalized way.

Interval<T>

Interval<T> type represents a continuous range of values. For example, a human age interval for teenagers can be defined as:

using Gapotchenko.FX.Math.Intervals;

// Define an inclusive interval of years ranging from 13 to 19.
// Inclusive interval has both its ends included in the interval.
// The formal interval representation is [13,19],
// where '[' denotes the start of the interval (inclusive),
// and ']' denotes the end of the interval (inclusive).
var teenagers = Interval.Inclusive(13, 19);

Given that range, it is now possible to do various operations:

Console.Write("Enter your age: ");
var age = int.Parse(Console.ReadLine());

// Everyone of an age between 13 and 19 years (inclusive) is a teenager.
// Using the interval notation, this can be stated as: age ∈ [13,19],
// where '∈' symbol denotes the "is an element of" operation.
if (teenagers.Contains(age))
    Console.WriteLine("Congrats, you are a teenager.");
else
    Console.WriteLine("Congrats, you are not a teenager.");

In this simple example, the value is just a number but it can be any comparable type. For example, it can be a System.Version, a System.DateTime, etc.

Interval Operations

The real power of intervals comes when you need to perform certain operations on them.

Overlap Detection

Interval<T>.Overlaps function returns a Boolean value indicating whether a specified interval overlaps with another:

var teenagers = Interval.Inclusive(13, 19);

// Adults are people of 18 years or older
// (the exact age of adulthood depends on a jurisdiction but we use 18 for simplicity).
// Using the interval notation, this is [18,∞),
// where ')' denotes the end of the interval (non-inclusive),
var adults = Interval.FromInclusive(18);

Console.Write(
    "Can teenagers be adults? The answer is {0}."
    teenagers.Overlaps(adults) ? "yes" : "no");

The snippet produces the following output:

Can teenagers be adults? The answer is yes.

Intersection

The intersection of two intervals returns an interval which has a range shared by both of them:

var teenagers = Interval.Inclusive(13, 19);
var adults = Interval.FromInclusive(18);

Console.WriteLine(
    "Adult teenagers have an age of {0}",
    teenagers.Intersect(adults));

The snippet produces the following output:

Adult teenagers have an age of [18,19].

Union

The union of two continuous intervals has the range that covers both of them:

var teenagers = Interval.Inclusive(13, 19);
var adults = Interval.FromInclusive(18);

Console.WriteLine(
    "Adults and teenagers have an age of {0}",
    teenagers.Union(adults));

The snippet produces the following output:

Adults and teenagers have an age of [13,inf).

Note the [13,inf) interval string in the output above. This is the ASCII variant of a formal [13,∞) notation. The ASCII notation is produced by Interval<T>.ToString() method by default. If you want the formal Unicode notation, you can pass U format specifier as in Interval<T>.ToString("U") method call:

Console.WriteLine(
    "Adults and teenagers have an age of {0:U}",
    teenagers.Union(adults));

which produces the output using Unicode mathematical symbols:

Adults and teenagers have an age of [13,∞).

Interval Construction

To define an interval, you can use a set of predefined methods provided by the static Interval type:

// [10,20]
interval = Interval.Inclusive(10, 20);

// (10,20)
interval = Interval.Exclusive(10, 20);

// [10,20)
interval = Interval.InclusiveExclusive(10, 20);

// (10,20]
interval = Interval.ExclusiveInclusive(10, 20);

// [10,∞)
interval = Interval.FromInclusive(10);

// (10,∞)
interval = Interval.FromExclusive(10);

// (-∞,10]
interval = Interval.ToInclusive(10);

// (-∞,10)
interval = Interval.ToExclusive(10);

Or you can explicitly construct an interval by using an Interval<T> constructor using the notion of boundaries:

// [10,20)
interval = new Interval<int>(IntervalBoundary.Inclusive(10), IntervalBoundary.Exclusive(20));

// (10,∞)
interval = new Interval<int>(IntervalBoundary.Exclusive(10), IntervalBoundary.PositiveInfinity<int>());

Special Intervals

There are a few special intervals readily available for use:

// An empty interval ∅
interval = Interval.Empty<T>();

// An infinite interval (-∞,∞)
interval = Interval.Infinite<T>();

// A degenerate interval [x;x]
interval = Interval.Degenerate(x);

ValueInterval<T>

ValueInterval<T> type provides a similar functionality to Interval<T> but it is a structure in terms of .NET type system, while Interval<T> is a class. The difference is that ValueInterval<T> can be allocated on stack without involving expensive GC memory operations, also it has tinier memory footprint.

All in all, ValueInterval<T> is the preferred interval type to use. Being totally transparent and interchangeable with Interval<T>, it comes with certain restrictions. For example, ValueInterval<T> cannot use a custom System.IComparer<T>, and thus it requires T type to always implement System.IComparable<T> interface. This is not an obstacle for most specializing types, but this is a formal restriction that may affect your choice in favor of Interval<T>.

Another scenario where you may prefer Interval<T> type better is when you need to pass it as a reference to many places in code. This may save some CPU time and memory in cases where T type is sufficiently large because passing the interval by reference avoids copying.

Interval Parsing

Intervals can be parsed from string representations using the standard interval notation. The parser supports both Unicode and ASCII representations of intervals.

Basic Parsing

Use Interval.Parse<T> to parse an interval from a string:

// Parse an inclusive interval [10,20]
var interval = Interval.Parse<int>("[10,20]");

// Parse an exclusive interval (5,15)
interval = Interval.Parse<int>("(5,15)");

// Parse a mixed interval [10,20)
interval = Interval.Parse<int>("[10,20)");

The parser supports both comma (,) and semicolon (;) as separators:

// Both forms are equivalent
interval = Interval.Parse<int>("[10,20]");
interval = Interval.Parse<int>("[10;20]");

Infinity

Infinite boundaries are supported using Unicode or ASCII notation:

// [10,∞) - Unicode infinity symbol
interval = Interval.Parse<int>("[10,∞)");

// [10,inf) - ASCII infinity notation
interval = Interval.Parse<int>("[10,inf)");

// (-∞,10] - Negative infinity
interval = Interval.Parse<int>("(-∞,10]");
interval = Interval.Parse<int>("(-inf,10]");

Empty and Degenerate Intervals

Empty intervals can be parsed using the empty set notation:

// Empty interval ∅
interval = Interval.Parse<int>("∅");
interval = Interval.Parse<int>("{}");

Degenerate intervals (single point) can be also parsed using set notation:

// Degenerate interval {5} which represents [5,5]
interval = Interval.Parse<int>("{5}");

TryParse

For scenarios where parsing might fail, use TryParse to avoid exceptions:

if (Interval.TryParse<int>("[10,20]", out var interval))
    Console.WriteLine($"Parsed interval: {interval}");
else
    Console.WriteLine("Failed to parse interval");

Culture-Specific Parsing

The parser respects culture-specific formatting when converting boundary values:

using System.Globalization;

// Parse with a specific culture
var culture = new CultureInfo("de-DE");
var interval = Interval.Parse<float>("[10,5;20,3]", culture);

ValueInterval Parsing

ValueInterval<T> also supports parsing with the same syntax:

var valueInterval = ValueInterval.Parse<int>("[10,20]");

Commonly Used Types

• Gapotchenko.FX.Math.Intervals.Interval<T>
• Gapotchenko.FX.Math.Intervals.ValueInterval<T>

Other Modules

Let's continue with a look at some other modules provided by Gapotchenko.FX:

• Gapotchenko.FX
• Gapotchenko.FX.AppModel.Information
• Gapotchenko.FX.Collections
• Gapotchenko.FX.Console
• Gapotchenko.FX.Data
• Gapotchenko.FX.Diagnostics
• Gapotchenko.FX.IO
• Gapotchenko.FX.Linq
• Gapotchenko.FX.Math
  • Gapotchenko.FX.Math.Combinatorics
  • Gapotchenko.FX.Math.Graphs
  • ➴ Gapotchenko.FX.Math.Intervals
  • Gapotchenko.FX.Math.Metrics
• Gapotchenko.FX.Memory
• Gapotchenko.FX.Security.Cryptography
• Gapotchenko.FX.Text
• Gapotchenko.FX.Threading
• Gapotchenko.FX.Tuples
• Gapotchenko.FX.Versioning

Or look at the full list of modules.