Curves & Surfaces - landscape element design , surface definition and milling paths

Curves & Surfaces landscape element,

surface definition and milling paths

Mahshid Moghadasi MDC ‘21 Cornell University

SURFACE GENERATION EXPLORATION

Surface 01

No function

Function 01

Function 02

Function 03

Surface 02

Surface 03

Surface 04

Surface 05

Function 04

Function 05

Function 06

Surface Generation Codes Surface 02

Surface 03

List < Point3d > points = new List<Point3d>(); int columns = 50; int rows = 50; double stepX = (regionP2.X - regionP1.X) / (columns - 1); double stepY = (regionP2.Y - regionP1.Y) / (rows - 1);

List < Point3d > points = new List<Point3d>(); int columns = 50; int rows = 50; double stepX = (regionP2.X - regionP1.X) / (columns - 1); double stepY = (regionP2.Y - regionP1.Y) / (rows - 1);

double x = regionP1.X; double y = regionP1.Y; double z = 0.0;

double x = regionP1.X; double y = regionP1.Y; double z = 0.0;

for (int j = 0; j < rows; ++j) { x = regionP1.X; for (int i = 0; i < columns; ++i){

for (int j = 0; j < rows; ++j) { x = regionP1.X; for (int i = 0; i < columns; ++i){

}

z = 1.5 * Math.Cos(x * 0.5); Point3d p = new Point3d(x, y, z);

z = 2 * Math.Cos(x * 0.2) * Math.Sin(y * 0.4); Point3d p = new Point3d(x, y, z);

points.Add(p); x += stepX;

points.Add(p); x += stepX;

} y += stepY;

Surface srf = NurbsSurface.CreateThroughPoints(points, columns, rows, 3, 3, false, false);

Surface 04

}

} y += stepY;

Surface srf = NurbsSurface.CreateThroughPoints(points, columns, rows, 3, 3, false, false);

Surface 05

List < Point3d > points = new List<Point3d>(); int columns = 50; int rows = 50; double stepX = (regionP2.X - regionP1.X) / (columns - 1); double stepY = (regionP2.Y - regionP1.Y) / (rows - 1);

List < Point3d > points = new List<Point3d>(); int columns = 50; int rows = 50; double stepX = (regionP2.X - regionP1.X) / (columns - 1); double stepY = (regionP2.Y - regionP1.Y) / (rows - 1);

double x = regionP1.X; double y = regionP1.Y; double z = 0.0;

double x = regionP1.X; double y = regionP1.Y; double z = 0.0;

for (int j = 0; j < rows; ++j) { x = regionP1.X; for (int i = 0; i < columns; ++i){

for (int j = 0; j < rows; ++j) { x = regionP1.X; for (int i = 0; i < columns; ++i){

}

z = Math.Sin((x - 15) * (y - 15) * 0.1); Point3d p = new Point3d(x, y, z);

z = 0.25 * Math.Abs(x) * Math.Pow(Math.Cos(0.5 * y), 2); Point3d p = new Point3d(x, y, z);

points.Add(p); x += stepX;

points.Add(p); x += stepX;

} y += stepY;

Surface srf = NurbsSurface.CreateThroughPoints(points, columns, rows, 3, 3, false, false);

}

} y += stepY;

Surface srf = NurbsSurface.CreateThroughPoints(points, columns, rows, 3, 3, false, false);

Functionsâ&#x20AC;&#x2122; Codes

Function 01 public double changeZNClosed (Point3d p, Curve crv, double a, double b){ double t = 0.0; crv.ClosestPoint(p, out t); Point3d cp = crv.PointAt(t); double d = cp.DistanceTo(p); double nt = crv.Domain.NormalizedParameterAt(t); nt = mapFromUnit(nt, 0, Math.PI); double change = Math.Sin(nt) * a * Math.Exp(-b * d * d); return change; }

Function 02 public double changeZNormalized (Point3d p, Curve crv, double a, double b){ double t = 0.0; crv.ClosestPoint(p, out t); Point3d cp = crv.PointAt(t); double d = cp.DistanceTo(p); double nt = crv.Domain.NormalizedParameterAt(t);

}

double change = nt * a * Math.Exp(-b * d * d); return change;

Function 03 public double changeZ (Point3d p, Curve crv, double a, double b){ double t = 0.0; crv.ClosestPoint(p, out t); Point3d cp = crv.PointAt(t); double d = cp.DistanceTo(p); double change = a * Math.Exp(-b * d * d); return change; }

Function 04 public double changeZFallSin (Point3d p, Curve crv, double a, double b){ double t = 0.0; crv.ClosestPoint(p, out t); Point3d cp = crv.PointAt(t); double d = cp.DistanceTo(p); double nt = crv.Domain.NormalizedParameterAt(t); double change = nt * Math.Sin(Math.PI * nt * 3) * Math.Exp(-b * d * d) * 8; return change; }

Function 05 public double changeZSin (Point3d p, Curve crv, double a, double b, double c){ double t = 0.0; crv.ClosestPoint(p, out t); Point3d cp = crv.PointAt(t); double d = cp.DistanceTo(p); double nt = crv.Domain.NormalizedParameterAt(t); double change = 2 * Math.Sin(Math.PI * d * c); return change; }

Function 06 public double changeZSinCos (Point3d p, Curve crv, double a, double b, double c){ double t = 0.0; crv.ClosestPoint(p, out t); Point3d cp = crv.PointAt(t); double d = cp.DistanceTo(p); double nt = crv.Domain.NormalizedParameterAt(t);

}

double change = 2 * Math.Exp(-b * Math.Cos(Math.PI * d * d * 0.1) * Math.Cos(Math.PI * d * d * 0.1)); return change;

STREAMLINE FORCES EXPLORATION

Point Attraction

Point Attraction + Contour

Point Attraction + Rotation

Point Attraction + Slope

Magnet

Magnet + Contour

Magnet + Rotation

Magnet + Slope

Slope

Slope + Contour

Slope + Rotation

All Attracting Magnet

Contour

Rotation + Contour

Rotation

All Repellent Magnet

Streamline Forces Codes

Attraction public Vector3d attractionVec(Surface s, double u, double v, Point3d p, double strength){ Point3d uvp = s.PointAt(u, v); Vector3d directionv = p - uvp; directionv.Unitize(); Vector3d nv = s.NormalAt(u, v); double dotproduct = directionv * nv; Vector3d projected = dotproduct * nv; Vector3d attv = directionv - projected; attv.Unitize(); return (attv * strength); }

Magnetic Force public Vector3d magnetVec (Surface s, double u, double v, List<Point3d> pos, List<Point3d> neg, double strength){ Point3d uvp = s.PointAt(u, v); Vector3d magv = Vector3d.Zero; foreach (Point3d p in pos){ Vector3d df = p - uvp; double L = df.Length; magv -= df * (1.0 / (L * L)); } foreach (Point3d p in neg){ Vector3d df = p - uvp; double L = df.Length; magv += df * (1.0 / (L * L)); } }

return (magv * 1 / (strength + 0.00001));

Rotation public Vector3d rotationVec(Surface s, double u0, double v0, List<Point3d> rps, double strength){ Vector3d fv = Vector3d.Zero; Point3d uvp = s.PointAt(u0, v0); foreach (Point3d p in rps){ Vector3d directionv = p - uvp; double L = directionv.Length; fv += directionv * (1.0 / (L * L )); } Vector3d nv = s.NormalAt(u0, v0); Vector3d v = Vector3d.CrossProduct(fv, nv); return (v * strength); }

Streamline Forces Codes

Contour public Vector3d contourVec(Surface s, double u, double v, double strength){ Point3d p = new Point3d(0.0, 0.0, 0.0); Point3d uvp = s.PointAt(u, v); Vector3d directionv = p - uvp; directionv.Unitize(); Vector3d nv = s.NormalAt(u, v); return (Vector3d.CrossProduct(nv, new Vector3d(0, 0, 1)) * strength); }

Slope public Vector3d slopeVec(Surface s, double u, double v, double strength){ Point3d uvp = s.PointAt(u, v); Vector3d n = s.NormalAt(u, v); Vector3d slope = new Vector3d(n.X * n.Z, n.Y * n.Z, -n.X * n.X - n.Y * n.Y); double L = slope.Length; slope = slope * (1.0 / L * L); return (slope * strength); }

Streamline Creation //......................Creating Final Vector at each Point.............. public Vector3d fieldAt(Surface s, double u, double v, List<Point3d> pos, List<Point3d> neg, Point3d attPoint, List<Point3d> rotationPoint, double magnetWeight, double attractionWeight, double rotationWeight, double contourWeight, double slopeWeight) { Vector3d Vector3d Vector3d Vector3d Vector3d

}

fv rv cv mv sp

= = = = =

attractionVec(s, u, v, attPoint, attractionWeight); rotationVec(s, u, v, rotationPoint, rotationWeight); contourVec(s, u, v, contourWeight); magnetVec(s, u, v, pos, neg, magnetWeight); slopeVec(s, u, v, slopeWeight);

Vector3d sumv = fv + rv + cv + mv + sp; //All are projected on surface--> result is also on surface sumv.Unitize(); sumv *= 0.2; return sumv;

//........................Creating the streamlines for milling............. public Polyline createStreamline (Surface s, Point3d seedPoint, int n, List<Point3d> pos, List<Point3d> neg, Point3d attPoint, List<Point3d> rotationPoint, double magnetWeight, double attractionWeight, double rotationWeight, double contourWeight, double slopeWeight) { Polyline pl = new Polyline();

}

Point3d p = seedPoint; for (int i = 0; i < n; ++i){ double u = 0.0; double v = 0.0; s.ClosestPoint(p, out u, out v); p = s.PointAt(u, v); pl.Add(p); Vector3d move = fieldAt(s, u, v, pos, neg, attPoint, rotationPoint, magnetWeight, attractionWeight, rotationWeight, contourWeight, slopeWeight); p = p - move; } return pl;

FINAL DESIGN

1) z = 2 * Math.Cos(x * 0.4) * Math.Sin(y * 0.4);

2) z = 2 * Math.Cos(x * 0.4) * Math.Sin(y * 0.4) + y * 0.2;

3) p.Z -= changeZSinCos2(p, crv1, a, b, c);

Hydrophobic tree on the heights

4) streamlines show the water flow Hydrophilic trees in the valleys

//...................Surface .....................

List < Point3d > points = new List<Point3d>(); int columns = 50; int rows = 50; double stepX = (regionP2.X - regionP1.X) / (columns - 1); double stepY = (regionP2.Y - regionP1.Y) / (rows - 1); double x = regionP1.X; double y = regionP1.Y; double z = 0.0; for (int j = 0; j < rows; ++j){ x = regionP1.X; for (int i = 0; i < columns; ++i){ z = 2 * Math.Cos(x * 0.4) * Math.Sin(y * 0.4) + y * 0.2; Point3d p = new Point3d(x, y, z); p.Z -= changeZSinCos2(p, crv1, a, b, c); points.Add(p); x += stepX;

} y += stepY;

} Surface srf = NurbsSurface.CreateThroughPoints(points, columns, rows, 3, 3, false, false); List <Point3d> uvpoints = new List<Point3d>(); List <Vector3d> fvectors = new List<Vector3d>(); Vector3d z = new Vector3d(0, 0, 1); //..............Points at UVs...................... int nx = 60; int ny = 60; double stepnx = (srf.Domain(0)[1] - srf.Domain(0)[0]) / (nx - 1); double stepny = (srf.Domain(1)[1] - srf.Domain(1)[0]) / (ny - 1); for (int j = 0; j < ny; ++j){ for (int i = 0; i < nx; ++i){ double u = i * stepnx + srf.Domain(0)[0]; double v = j * stepny + srf.Domain(1)[0]; Point3d uvp = srf.PointAt(u, v); uvpoints.Add(uvp); //..............Field Vector at UVs............

Vector3d sumv = fieldAt(srf, u, v, posPoints, negPoints, attPoint, rotationPoints, magnetWeight, attractionWeight, rotationWeight, contourWeight,

slopeWeight); fvectors.Add(sumv); //......tangents................

Vector3d[] dv; //both tangents srf.Evaluate(u, v, 1, out uvp, out dv); tangents0.Add(dv[0] * 0.5); tangents1.Add(dv[1] * 0.5);

} } //..............Stream Lines......................

List <Polyline> slines = new List<Polyline>(); foreach (Point3d p in seedPoints){ Polyline pl = createStreamline(srf, p, n, posPoints, negPoints, attPoint, rotationPoints, magnetWeight, attractionWeight, rotationWeight, contourWeight,

slopeWeight); slines.Add(pl); }

FINAL RESULT Magnetic Force + Slope Vectors Streamlines focus on the hydrophilic trees and get away from from the hydrophobic ones, showing the water flow.

Front View

Perspective View

Plan View