- /
-
Trefoil Spring
on 3 Nov 2024
- 74
- 568
- 0
- 1
- 1999
Cite your audio source here (if applicable):
drawframe(1);
Write your drawframe function below
function drawframe(f)
% draws the surface of a 'spring' stretched along the path of a trefoil
% knot with a compression wave travelling through it
persistent N n H t x y z X Y Z u v w R nT F CM U MC L C S D p
% setup
if f == 1
%% helper functions
F = @cellfun;
CM = @cell2mat;
MC = @mat2cell;
L = @linspace;
C = @cos;
S = @sin;
U = 'UniformOutput';
D = {'XData','YData','ZData','CData'};
p = 2*pi;
%% figure/axes
% set axes props
set(gca,'Visible','off','Units','Normalized','InnerPosition',[0 0 1 1],'XLim',[-4 4],'YLim',[-4.4 3.6],'ZLim',[-4 4],'NextPlot','Add');
pbaspect([1 1 1]);daspect([1 1 1]);
% set figure background color
set(gcf,'Color',[0 0 0]);
% lighting
lighting gouraud;
camlight headlight;
%% tube/spiral properties
% number of circles used to define the tube surface
N = 6720;
% number of points to create each circle cross-section
n = 50;
% radius of the central rube, defines the distance of the spiral
% from the central path
R = .5;
% Number of spiral turns around the central tube
nT = 48;
% parameter for the trefoil knot curve
t = L(0, p, N);
% trefoil knot parametric equations
x = S(t) + 2 * S(2 * t);
y = C(t) - 2 * C(2 * t);
z = -S(3 * t);
% tube surface
H = surf(D{1},[],D{2},[],D{3},[],D{4},[],'EdgeColor','none','FaceAlpha',1);
end
%% set up compression wave
% index shift for this frame
I = mod((f - 1) * (N / 96), N); % Shift by fixed amount per frame
% modify t with cosine shaped compression effect, centered on mod((f-1)/96,95)*2*pi, intensity 0.4
tM = cumsum(1-.4*C(t-mod((f-1)/96,95)*p));
tM = tM / max(tM) * p;
% shift the compressed t
tS = [I+1:N,1:I];
tSh = tM(tS);
% interpolate x, y, and z using the compressed, shifted t values
Q = @(a) interp1(t,a(tS),tSh,'pchip');
xC = Q(x); yC = Q(y); zC = Q(z);
%% create the tube and the spiral
% anonymous func to set last column equal to first column
W = @(a) a(:,[1:N-1,1]);
% get trefoil tube surface (X,Y,Z) and spiral path (u,v,w) coordinates
[X,Y,Z,u,v,w] = ST(xC,yC,zC,R);
% match spiral endpoints
u = W(u); v = W(v); w = W(w);
% create another tube that follows the spiral path
[X,Y,Z,~,~,~] = ST(u,v,w,.1);
% match endpoints
X = W(X); Y = W(Y); Z = W(Z);
% reorder the circle points to correct overtwisting
% -> for each circle, i, rotate circle i+1 until distance between
% points minimized
for i = 1:N-1
j = i+1;
[~,l] = min(sqrt((X(:,j)-X(1,i)).^2+(Y(:,j)-Y(1,i)).^2+(Z(:,j)-Z(1,i)).^2));
k = [l:n,1:l-1];
X(:,j) = X(k,j); Y(:,j) = Y(k,j); Z(:,j) = Z(k,j);
end
% sometimes need the swap below if the spiral tube radius is very small
% X(:,1) = X(:,end);
% Y(:,1) = Y(:,end);
% Z(:,1) = Z(:,end);
% make sure the last point of each circle is identical to the first point
X(n,:) = X(1,:);
Y(n,:) = Y(1,:);
Z(n,:) = Z(1,:);
%% update surface
% update tube surface coordinates
set(H,D{1},X,D{2},Y,D{3},Z,D{4},repmat(L(0,1,N),n,1));
% shift the colormap to move colors along the spiral
M = hsv;
Ci = round((f/96)*256);
colormap(M([Ci+1:end,1:Ci],:));
drawnow
function [TX,TY,TZ,SX,SY,SZ] = ST(PX,PY,PZ,CR)
% returns surface coordinates for a tube (TX, TY, & TZ) traveling along the path
% defined by PX, PY, and PZ. Also returns coordinates for the path of a
% spiral (SX, SY, & SZ) traveling along the outside of the tube.
c = @cross;
g = @norm;
d = @diff;
% points along the path
P = [PX',PY',PZ'];
% cell array of normalized tangent vectors at each point along the path
tN = F(@(a) a/g(a),MC([d(P([1:N,1],1)) d(P([1:N,2],2)) d(P([1:N,3],3))],ones(N,1),3),U,0);
% find two vectors normal to each tangent to form a basis for each circle
N1 = F(@(a) c(a,[1 0 0]),tN,U,0);
N1 = F(@(a) a/g(a),N1,U,0);
N2 = F(@(a,b) c(a,b),tN,N1,U,0);
% generate circle points around each point on the path
T = L(0, p, n);
% coordinates for each circle
cPt = F(@(a,b) CR*(a'*C(T)+b'*S(T))',N1,N2,U,0);
% make sure the endpoints exactly match
cPt = F(@(a) a([1:end-1,1],:),cPt,U,0);
% cell array of tube surface points
K = F(@(a,b) a+b,MC(P,ones(N,1),3),cPt,U,0);
% matrices of tube surface points
TX = CM(F(@(a) a(:,1),K,U,0)');
TY = CM(F(@(a) a(:,2),K,U,0)');
TZ = CM(F(@(a) a(:,3),K,U,0)');
% spiral angles
A = p * nT * (1:N) / (N-1);
% spiral points
SP = CM(F(@(a,b,i) [P(i,1)+CR*C(A(i))*a(1)+CR*S(A(i))*b(1);P(i,2)+CR*C(A(i))*a(2)+CR*S(A(i))*b(2);P(i,3)+CR*C(A(i))*a(3)+CR*S(A(i))*b(3)],N1',N2',arrayfun(@(i) i,1:N,U,0),U,0));
SX = SP(1,:);
SY = SP(2,:);
SZ = SP(3,:);
end
end
Movie
Audio
This submission does not have audio.
