Module

Data.Geometria

Package
purescript-geometria
Repository
Ebmtranceboy/purescript-geometria

Re-exports from Data.Geometria.Ellipse

#Ellipse Source

newtype Ellipse
 --                                                        -- 
| ;TLDR :                                                    |
| ------                                                     |
|                                                            |
| e@                                                         |
|   (                                                        |
|     Ellipse                                                |
|       { center                                             |
|       , a                                                  |
|       , b                                                  |
|       , c                                                  |
|       }                                                    |
|   ) = ellipse center (radiusMax /\ radiusMin) alpha        |
|                                                            |
| represents the set of points (x/\y) satisfying:            |
|                                                            |
| a * (x - center.x)^2                                       |
| + b * (y - center.y)^2                                     |
| + c * (x - center.x) (y - center.y) - residualConstant(e)  |
| = 0                                                        |
|                                                            |
|                                                            |
 --                                                        --

ELLIPSES

Implicit ellipse representations: 5 degrees of freedom

Level-set representation:

f(x,y) = ax2 + by2 + cxy + dx + ey = 1

Matrix equation representation:

              _           _    _ _
             | a   c/2 d/2 |  | x |
     _    _  | c/2 b   e/2 |  | y |
    | x y 1| | d/2 e/2 -1  |  | 1 | = X^TAX = 0
     -    -   -           -    - -

Centering (abandonment of degree-1 terms):

Changes of variables: x = X+h and y = Y+k such that f(x,y) = g(X,Y) = aX2 + bY2 + cXY + L => [h; k] = [ec-2bd; cd-2ae] / (4ab-c2) and L = L(a,b,c,d,e,h,k) = ah2 + bk2 + chk + dh + ek. Furthermore, f(x,y) = 1 => aX2 + bY2 + cXY = l = 1-L. Then, ellipse representation can simply be:

              _       _    _ _
     _   _   | a   c/2 |  | x |
    | x y |  | c/2 b   |  | y | = X^TBX = l
     -   -    -       -    - -

(cf. ellipseInternal and residualConstant functions)

Alinement (coupling term abandonment, after centering):

B diagonalization, with B symmetric matrix:

  • one diagonal matrix D =[D00, 0; 0, D11] : 2 degrees of freedom
  • one angle t : 1 degree of freedom => thus 5 degree of freedom with the 2 ones from centering.

(cf. unCouple function)

Ellipses explicit representations:

Every point (x,y) of the unit circle is associated with a point (X,Y)from the (h,k)-centered ellipse thanks to the affine transformation:

 _ _   _ _   _              _  _                                 _  _ _
| X | | h | | cos(t) -sin(t) || s0=sqrt(|l/D00|) 0                || x |
| Y |=| k |+| sin(t)  cos(t) || 0                s1=sqrt(|l/D11|) || y |
 - -   - -   -              -  -                                 -  - -

whose linear part is like an SVD decomposition (if, as it has been done here, V is chosen as the identity, meaning that the four cardinal points from the unit circle correspond to the extreme points of the ellipse). This way, reciprocally, t, s0 and s1 parameters can be recovered, for every T = USV' matrix, by

  • diagonalizing T'T
  • orthonormalizing its eigen vectors
  • taking the square root of its eigen values.

(cf. ellipseDimensionsand fromUnitCircle functions)

Constructors

Instances

#unCouple Source

unCouple :: Ellipse -> Number /\ Number /\ Number

Equivalent to the diagonalization of the symmetric input matrix : t est l'angle de rotation autour de l'origine tel que c'=0 dans la nouvelle representation:

  • tan(2t) = c/(a-b)
  • vec = matrice de rotation d'angle t
  • val = recip vec * m * vec = [d00, 0; 0, d11]

#turnEllipse Source

turnEllipse :: Number -> Ellipse -> Ellipse

Rotation adapted to the chosen representation of an ellipse.

#steiner Source

steiner :: Point 2 -> Point 2 -> Point 2 -> Ellipse

Biggest ellipse in a triangle. Could also be defined with 5 points :

  • the 3 segment's middles,
  • the intersection of a median and the symmetric of the opposite side with respect to isobarycenter and
  • another intersection of a median and the symmetric of the opposite
    side with respect to isobarycenter.

#rytz Source

rytz :: Point 2 -> Point 2 -> Point 2 -> Ellipse

Biggest ellipse in a parallelogram defined by:

  • its center
  • the middle of one of its sides
  • the middle of the next side.

#residualConstant Source

residualConstant :: Ellipse -> Number

Value of the second member in the updated equation after centering.

#quadratic Source

quadratic :: Array (Point 2) -> Ellipse

Extracts the relevant parts of an ellipse defined by 5 points (all different from the origin): symmetric 2X2 matrix /\ position of the center

Computation of the parameters of an ellipse passing through the origin (point zero) CANNOT be done with the quadratic function even if the five points are different from the origin. See in the test file how to make two translations to use it anyway.

#moveEllipse Source

moveEllipse :: Vector 2 -> Ellipse -> Ellipse

Translation adapted to the chosen representation of an ellipse.

#fromUnitCircle Source

fromUnitCircle :: Ellipse -> Point 2 -> Point 2

Explicit construction of a point of the ellipse from a point of the unit circle such that the four cardinal circular points correspond to the extremities of the axes of the ellipse.

#foci Source

foci :: Ellipse -> (Point 2) /\ (Point 2)

The 2 points F0 and F1 such that MF0 + MF1 = 2 * max(ellipseDimensions e) for every M of the ellipse.

#expandEllipse Source

expandEllipse :: Number /\ Number -> Ellipse -> Ellipse

Scaling adapted to the chosen representation of an ellipse.

#ellipseDimensions Source

ellipseDimensions :: Ellipse -> Number /\ Number

Semi-axes of the ellipse

#ellipseCenter Source

ellipseCenter :: Ellipse -> Point 2

#ellipse Source

ellipse :: Point 2 -> Number /\ Number -> Number -> Ellipse

Smart constructor computing the ellipse parameters from :

  • its center
  • the values of the two semi-axes
  • the value of the tilt

#cardinal Source

cardinal :: Point 2 -> Point 2 -> Number -> Ellipse

Computes the relevant parts of an ellipse defined by :

  • its center
  • one of its cardinal points, (this sets the first semi-axis), and
  • the length of the other semi-axis. Could also be defined with 5 points :
  • the 4 cardinal points,
  • a fifth point defined as the sum of two consecutive cardinal vectors, over sqrt(2).

#brianchon Source

brianchon :: Point 2 -> Point 2 -> Point 2 -> Point 2 -> Point 2 -> Ellipse

The only ellipse tangent to the 5 sides of the polygon described by its ordered vertices.

Re-exports from Data.Geometria.Space

#revolution Source

revolution :: Vector 3 -> Number -> Vector 3 -> Vector 3

revolution n a u is the 3D-rotation of u of angle a around the normalized axis n.

#landing Source

landing :: Vector 3 -> Matrix Number

Matrix used by the land function.

#land Source

land :: Vector 3 -> Vector 3 -> Vector 3

land n is a 3D-rotation such that, for every 3D vector u in the plane P of normal vector n, land n u has a constant altitude (z-coordinate). Thanks to land, every computation on P can be performed in 2D. The rotation matrix is obtained by the landing function, and, as it is orthogonal, the inverse transformation is simply done by the transposed matrix.

Re-exports from Data.Geometria.Types

#Vector Source

newtype Vector :: Int -> Typenewtype Vector (n :: Int)

Constructors

Instances

#System Source

type System :: Int -> Typetype System (n :: Int) = Polynomial (Polynomial Number)

#Point Source

newtype Point :: Int -> Typenewtype Point (n :: Int)

Constructors

Instances

#Analytic Source

class Analytic a  where

Members

Instances

#EuclideanSpace Source

class EuclideanSpace a  where

Members

  • dot :: a -> a -> Number

    Scalar product

  • normalTo :: Array a -> a

    Builds the n-dimensioned vector needed for the provided array of (n-1) n-dimensioned independant vectors to be a R^n basis.

Instances

#Metric Source

class Metric a  where

Members

Instances

#Shape Source

class Shape :: Int -> (Int -> Type) -> Constraintclass Shape n s  where

Members

Instances

#vector Source

vector :: forall n. Point n -> Point n -> Vector n

#translatedBy Source

translatedBy :: forall n. Point n -> Vector n -> Point n

#system Source

system :: forall n s. ToString n s => IsSymbol s => Line n -> System n

Builds the (n-1) equations needed to describe a line of n-dimensioned points.

#segment Source

segment :: forall n. Point n -> Point n -> Segment n

#scale Source

scale :: forall n. Number -> Vector n -> Vector n

#rotated Source

rotated :: Number -> Vector 2 -> Vector 2

#projector Source

projector :: forall n s. ToString n s => IsSymbol s => Vector n -> Vector n -> Matrix Number

Matrix used by the project function.

#projection Source

projection :: forall n s. ToString n s => IsSymbol s => Vector n -> Vector n -> Vector n

projection d v projects a vector v on a vector d.

#project Source

project :: forall n s. ToString n s => IsSymbol s => Vector n -> Vector n -> Vector n -> Vector n

project n d u projects a vector u on a plane, passing through the origin and of normal vector n, using a direction parallel to d.

#point Source

point :: forall @n. Shape n Point => Polynomial Number -> Point n

#normalized Source

normalized :: forall n s. ToString n s => IsSymbol s => Vector n -> Vector n

#middle Source

middle :: forall n. Segment n -> Point n

#line Source

line :: forall n. Point n -> Point n -> Line n

#immerse Source

immerse :: forall a n n1. Analytic (a n) => Analytic (a n1) => Add n 1 n1 => a n -> a n1

Increments the dimension of a point/vector by adding a zero coordinate after the other coordinates.

#halfline Source

halfline :: forall n. Point n -> Vector n -> HalfLine n

#freeVector Source

freeVector :: forall @n. Shape n Vector => Polynomial Number -> Vector n

#drain Source

drain :: forall a s n n1. ToString n1 s => IsSymbol s => Analytic (a n) => Analytic (a n1) => Add n1 1 n => a n -> a n1

Decrement the dimension of a point/vector by removing its last coordinate.

#cosAngle Source

cosAngle :: forall n s. ToString n s => IsSymbol s => Vector n -> Vector n -> Number

#circle Source

circle :: Point 2 -> Number -> Circle

#anyVector Source

anyVector :: forall @n s. ToString n s => IsSymbol s => System n -> Vector n

#anyPoint Source

anyPoint :: forall n s. ToString n s => IsSymbol s => System n -> Point n

#(<+|) Source

Operator alias for Data.Geometria.Types.translatedBy (left-associative / precedence 6)

Modules
Data.Geometria
Data.Geometria.Ellipse
Data.Geometria.Space
Data.Geometria.Types