Module

Data.Geometria.Ellipse

Package: purescript-geometria
Repository: Ebmtranceboy/purescript-geometria

#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

Ellipse { a :: Number, b :: Number, c :: Number, center :: Point 2 }

Instances

#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

#ellipseCenter Source

ellipseCenter :: Ellipse -> Point 2

#ellipseInternal Source

ellipseInternal :: Ellipse -> Polynomial (Polynomial Number)

#residualConstant Source

residualConstant :: Ellipse -> Number

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

#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]

#ellipseDimensions Source

ellipseDimensions :: Ellipse -> Number /\ Number

Semi-axes of the 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.

#mkMatrix Source

mkMatrix :: Array (Point 2) -> Matrix Number

#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.

#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.

#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).

#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.

#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.

#moveEllipse Source

moveEllipse :: Vector 2 -> Ellipse -> Ellipse

Translation adapted to the chosen representation of an ellipse.

#turnEllipse Source

turnEllipse :: Number -> Ellipse -> Ellipse

Rotation adapted to the chosen representation of an ellipse.

#expandEllipse Source

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

Scaling adapted to the chosen representation of an ellipse.

#roots2 Source

roots2 :: Polynomial Number -> Array (Cartesian Number)

#roots4 Source

roots4 :: Polynomial Number -> Array (Cartesian Number)

#canonicalEllipseCircle Source

canonicalEllipseCircle :: Number -> Number -> Number -> Number -> Number -> Array (Point 2)

#canonicalEllipseLine Source

canonicalEllipseLine :: Number -> Number -> Number -> Number -> Number -> Array (Point 2)

Modules: Data.Geometria; Data.Geometria.Ellipse; Data.Geometria.Space; Data.Geometria.Types