basis.math

AffineSpace

trait AffineSpace extends AnyRef

An abstract principal homogeneous space over the additive group of a linear space. Vector addition acts freely and transitively over the point set. To the extent practicable, the following axioms should hold.

Axioms:

Source
AffineSpace.scala
Example: 

  1. // You can abstract over affine spaces by parameterizing a class or
// function with a subtype of AffineSpace with Singleton. Type elements
// with the #Point, #Vector and #Scalar type projections of your
// AffineSpace type parameter.
def testAffineSpaceOperations[A <: AffineSpace with Singleton](
    p: A#Point, q: A#Point, u: A#Vector, v: A#Vector): Unit = {
  assert((p + u) + v == p + (u + v), "associativity of point-vector addition")
  assert(p + (-v) == p - v, "existence of point-vector subtraction")
  assert((p + v) - (q + v) == p - q, "existence of point-point subtraction")
}

// Alternatively, functions can use path-dependent types of an AffineSpace parameter.
def testAffineSpaceIdentities(A: AffineSpace)(p: A.Point, q: A.Point, v: A.Vector): Unit = {
  import A._
  assert(p + zero == p, "existence of additive identity vector")
  if (p + v == q + v) assert(p == v, "uniqueness of points")
}
Version

0.1

Since

0.0

Linear Supertypes
AnyRef, Any
Known Subclasses
Integer, R2, R3, R4, RI2, RI3, Real, Z2, Z3, ZI2, ZI3
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Type Members

  1. abstract type Point <: PointElement

    The type of points in this affine space.

  2. trait PointElement extends Any

    A point in this affine space.

  3. type Scalar = Element

    The type of scalars in this affine space.

  4. type Vector = VectorSpace.Vector

    The type of vectors in this affine space.

Abstract Value Members

  1. abstract val Scalar: Ring

    Returns the scalar set of this affine space.

  2. abstract val Vector: VectorSpace

    Returns the vector space underlying this affine space.

  3. abstract def origin: Point

    Returns the origin of this affine space.

Concrete Value Members

  1. final def !=(arg0: Any): Boolean

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  2. final def ##(): Int

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  3. final def ==(arg0: Any): Boolean

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  4. final def asInstanceOf[T0]: T0

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  5. def clone(): AnyRef

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  6. final def eq(arg0: AnyRef): Boolean

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  7. def equals(arg0: Any): Boolean

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  8. def finalize(): Unit

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  9. final def getClass(): Class[_]

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  10. def hashCode(): Int

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  11. final def isInstanceOf[T0]: Boolean

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  12. final def ne(arg0: AnyRef): Boolean

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  13. final def notify(): Unit

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  14. final def notifyAll(): Unit

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  15. final def synchronized[T0](arg0: ⇒ T0): T0

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  16. def toString(): String

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  17. final def wait(): Unit

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  18. final def wait(arg0: Long, arg1: Int): Unit

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  19. final def wait(arg0: Long): Unit

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