clc
clear

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ME 4041 Homework 2
Date: 2/9/2012
Author: Harrison Jones
Notes: This homework is extremely tedious
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Question 1. Given a 2D parametric curve
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Question 2. Given a multi-segment 2D cubic Hermite curve.
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%DATA:
    %Point 0
    p0 = [1,5];
    p0u = [3,3];
    %Point 1
    p1 = [3,8];
    %Point 2
    p2 = [6,4];
    %Point 3
    p3 = [11,6];
    %Point 4
    p4 = [15,2];
    p4u = [3,-3];


%{
Question 2.1: Find the tangent vectors:
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% WORK - Finding the tangents
    p1u = -(45*p0+15*p0u-12*p1-42*p2+12*p3-3*p4+p4u)/56;
    p1uf = rats(p1u);

    p2u = (3*p2-4*p1u-p0u-3*p0);
    p2uf = rats(p2u);

    p3u = (3*p3-4*p2u-p1u-3*p1);
    p3uf = rats(p3u);

    p3ua = (3*p4-p4u-3*p2-p2u)/4
    p3uaf = rats(p3ua)
% Solutions
    %p1u = [1.9286 -1.2321] %[27/14 -69/56] in fractions
    %p2u = [4.2857 -1.0714] %[30/7 -15/14] in fractions
    %p3u = [4.9286 -0.4821] %[69/14 -27/56] in fractions

%{
Question 2.2: Find the algebraic form for each segment
%}
% WORK - For segment A ((1,5)->(3,8))
    %p(u) = Aa*u^3+Ab*u^2+Ac*u+Ad
    Aa = 2*p0-2*p1+p0u+p1u
    Ab = -3*p0+3*p1-2*p0u-p1u
    Ac = p0u
    Ad = p0
% Solutions
    %Aa = [0.9286 -4.2321]
    %Ab = [-1.9286 4.2321]
    %Ac = [3 3]
    %Ad = [1 5]
    %p(u)A = [0.9286 -4.2321]u^3 + [-1.9286 4.2321]u^2 + [3 3]u + [1 5]

% WORK - For segment B ((3,8)->(6,4))
    %p(u) = Ba*u^3+Bb*u^2+Bc*u+Bd
    Ba = 2*p1-2*p2+p1u+p2u
    Bb = -3*p1+3*p2-2*p1u-p2u
    Bc = p1u
    Bd = p1
% Solutions
    %Ba = [0.2143 5.6965]
    %Bb = [0.8571 -8.4644]
    %Bc = [1.9286 -1.2321]
    %Bd = [3 8]
    %p(u)B = [0.2143 5.6965]u^3 + [0.8571 -8.4644]u^2+[1.9286 -1.2321]u+[3 8]

% WORK - For segment C ((6,4)->(11,6))
    %p(u) = Ca*u^3+Cb*u^2+Cc*u+Cd
    Ca = 2*p2-2*p3+p2u+p3u
    Cb = -3*p2+3*p3-2*p2u-p3u
    Cc = p2u
    Cd = p2
% Solutions
    %Ca = [-0.7857 -5.5536]
    %Cb = [1.5 8.625]
    %Cc = [4.2857 -1.0714]
    %Cd = [6 4]
    %p(u)C = [-0.7857 -5.5536]u^3 + [1.5 8.625]u^2+[4.2857 -1.0714]u+[6 4]

% WORK - For segment D ((11,6)->(15,2))
    %p(u) = Da*u^3+Db*u^2+Dc*u+Dd
    Da = 2*p3-2*p4+p3u+p4u
    Db = -3*p3+3*p4-2*p3u-p4u
    Dc = p3u
    Dd = p3
% Solutions
    %Da = [-0.0714 4.5179]
    %Db = [-0.8571 -8.0357]
    %Dc = [4.9286 -0.4821]
    %Dd = [11 6]
    %p(u)D = [-0.0714 4.5179]u^3 + [-0.8571 -8.0357]u^2+[4.9286
    %-0.4821]u+[11 6]