Normals

We have seen in the previous post the patterns made by the the tangents of certain parametric lines. This is a follow-up on that post and we will look at the same closed figures by plotting normals to them.

======================================================================================================

Introduction:

Definition: The simplest description of a normal is that it is a line segment perpendicular to the curve or the surface at the point where it is rooted. This is the Wikipedia link.

Calculation:

If you have a two-dimensional curve defined by the parametric equations  x = x(t) and y = y(t)

The general equation for the normal at any point of such curve would be  Nx(t) = x”(t) and Ny(t) = y”(t)

Example:

  • Parametric Equation of a circle is             x = x(t) =  +Sin(t)   and         y = y(t) = +Cos(t);
  • Tangent at any point is                         Tx(t) = x'(t) = +Cos(t)   and    Ty(t) = y'(t) = -Sin(t);
  •  Normal at any point is                        Nx(t) = x”(t) = -Sin(t)    and    Ny(t) = y”(t) = -Cos(t);

======================================================================================================

Note 1   In this series, there will not be a separate image for the curve and the normals. The curve will be a closed line and it will be conspicuous enough to be visually separable from the other lines. 

Note 2 This is also the longest post till date of the fermibot blog. 

======================================================================================================

Three lobes

1. 400 Normals

3 Lobes 400

2. 1000 Normals

3 Lobes 1000 0.2

3. 4000 Normals

3 Lobes 4000 O 05

======================================================================================================

Four Lobes

1. 400 Normals

4 Lobes 200 O .2

2. 1000 Normals

4 Lobes 1000 O .2

3. 4000 Normals

4 Lobes 4000 O .05

======================================================================================================

Five Lobes

1. 400 Normals

5 Lobes 400 O .2

2. 1000 Normals

5 Lobes 1000 O .2

3. 4000 Normals

5 Lobes 4000 O .05

======================================================================================================

6 Lobes

In this case, the normals were too long. So, I have shortened them, to show the details of the center.

1. 400 Normals

6 Lobes 400 O .2

2. 1000 Normals

6 Lobes 1000 O .2

3. 4000 Normals

6 Lobes 4000 O .05

======================================================================================================

Bonus 1 (5 Lobes Version 2)

1. 200 Normals

5 Lobes Bonus 200 O .2

2. 400 Normals

5 Lobes Bonus 400 O .2

3. 4000 Normals

5 Lobes Bonus 4000 O .05

======================================================================================================

Bonus 2 (6 Lobes Version 2)

1. 200 Normals

6 Lobes Bonus 200 O .2

2. 400 Normals

6 Lobes Bonus 400 O .2

3. 4000 Normals

6 Lobes Bonus 4000 O .05

======================================================================================================

Bonus 3 (Bi-Color Flora)

  • For one curve, both the tangent and the Normals are shown at the same time
  • Notice the alignment and the striking symmetry of the objects. 

1. 200 Normals + 200 Tangents

5 Lobes Tang and Norm 200 O .3

2. 2000 Normals + 2000 Tangents

5 Lobes Tang and Norm 2000 + 2000 O .05

======================================================================================================

******************************************************************************************************

======================================================================================================

 

 

 

 

 

 

 

 

 

.

4 thoughts on “Normals

  1. Every weekend i used to pay a visit this web site, because i want enjoyment, for the reason that this this web site conations really nice funny data too.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s