How to make curve fitting by spline between two Arcs

If i would like to get spline equation the fits the formula below "cycloidal pinion" as i would like to make an arc in tip of the tooth how to make it?

% Define Parameters
pitchDiameter = 6; % Diameter of the rolling circle
rollerDiameter = 0.75; % Diameter of a smaller circle centered on the focal point of the rolling circle
noOfRollers = 6; % number of rollers (teeth) in pinion
noOfTeeth = 5; % desired number of teeth in rack assembly
webThickness = 2; % distance from center of roller at lowest point, and bottom edge of rack

% Calculated Constants
pi_val = pi;
rollerRadius = rollerDiameter / 2;
pitchRadius = pitchDiameter / 2;
pitchCircumference = pitchDiameter * pi_val;

% Cycloid Parametric Equations
theta = linspace(0, 2*pi_val, 100); % Create a range of angles
xValuesA = pitchRadius * (theta - sin(theta)); % X values for cycloid
yValuesA = pitchRadius * (1 - cos(theta)); % Y values for cycloid

% Offset Cycloid
offsetSecondCycloid = @(input) -input + (pitchCircumference / noOfRollers);
xValuesB = offsetSecondCycloid(xValuesA);
yValuesB = yValuesA;

% Plot Basic and Offset Cycloids
figure;
plot(xValuesA, yValuesA, '--', xValuesB, yValuesB, '--');
title('Basic and Offset Cycloid Curves');
xlabel('X');
ylabel('Y');
axis equal;
grid on;
hold on;

Answer 1

[William Rose]

@sila,

If you connect the two curves with a semicircle, then the first derivatives match at the connecting points. However, the radii of curvature do not match at the connecting points. See green curve below.

pitchRadius = 3; % radius of the rolling circle
noOfRollers = 6; % number of rollers (teeth) in pinion
% Cycloid Parametric Equations
theta = linspace(0, 2*pi, 100); % Create a range of angles
xValuesA = pitchRadius * (theta - sin(theta)); % X values for cycloid
yValuesA = pitchRadius * (1 - cos(theta)); % Y values for cycloid
% Offset Cycloid
offsetSecondCycloid = @(input) -input + pitchRadius*2*pi/noOfRollers;
xValuesB = offsetSecondCycloid(xValuesA);
yValuesB = yValuesA;
% Connectors
r1=pitchRadius*pi/noOfRollers;  % semicrcle radius
xConnCtr=r1;                    % x coordinate of semicircle center
xConn=xConnCtr+r1*cos(theta/2); % x coords of semicircle
yConn1=-r1*sin(theta/2);        % y coords of semicircle
% Plot Basic & Offset Cycloids & Connector
figure;
plot(xValuesA, yValuesA,'-r',xValuesB,yValuesB,'-b',xConn,yConn1,'-g');
title('Basic and Offset Cycloid Curves'); xlabel('X'); ylabel('Y');
legend('Basic','Offset','Semicircle'); 
axis equal; grid on; hold on

You do not use constants rollerDiameter, noOfTeeth, or webThickness in your calculations. Please explain what they represent - perhaps a diagram will make it clear. I do not understand what you mean by "i would like to make an arc in tip of the tooth how to make it".

A protracted, or prolate, cycloid is generated by

where R2>R1. When R2=R1, these equations generate a standard cycloid.

The prolate cycloid looks similar to the two segments you drew, but it has a smooth connecting portion. See plot below.

R1=3; R2=4; 
theta=-pi/2:pi/100:5*pi/2;
x1=R1*theta-R1*sin(theta);  % standard cycloid
y1=R1-R1*cos(theta);        % standard cycloid
x2=R1*theta-R2*sin(theta);  % prolate cycloid
y2=R1-R2*cos(theta);        % prolate cycloid
figure
plot(x1,y1,'-r',x2,y2,'-b');
grid on; axis equal; legend('Standard','Prolate'); title('Cycloids')