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Parameterization of a 3d body, surface and a curve

Akash Ray

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Chosen body descriptions

  • Cone
    • Height 10
    • Radius 5
  • Surface
    • Curved surface of the Cone
  • Curve
    • Curved bottom edge of the Cone

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3D Body parameterization

  • The parametric equation of the cone is similar to that of a cylinder. One would just have to vary the radius as one increases the height.

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Code to generate the cone

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Curve parameterization

  • The curve shown in red is essentially a line spiralling down the curved surface of the cone.

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Length of the curve

So from the integration, the length of the curve came out to be R*T, which is equal to 2*pi*R since T=2*pi. This is correct since the curved edge of the cone is basically a circle which has a circumference of 2*pi*R.

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Code to generate the Curve

No of segments

Approximate length from the code (La)

Exact length from the formula (2*pi*R)=(Le)

Relative percentage Error

(|La-Le|/Le)*100

10

30.902

31.416

1.636

100

31.410

31.416

0.019

1000

31.416

31.416

0

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Parametric equations for the Surface

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Area of the Surface

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Check for the correctness of the expression for the curved surface of the Cone

2*pi

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Code for the Surface

No of segments

Approximate area from the code (Aa)

Exact area from the formula (pi*R*l)=(Ae)

Relative percentage Error

(|Aa-Ae|/Ae)*100

10

153.979

175.620

12.323

100

173.818

175.620

1.026

1000

175.444

175.620

0.100