EMAT 6690

ESSAY 2 Site

Heart to Heart

**This essay concerns
the light "Hearted" subject of the equations assorted
with the heart shape. What will be shown here, is the following:**

Equation derived from an ellipse and its reflection across the y-axis.A little explanation as to why this particular orientation of an ellipse seems to work.A brief set of examples as to effect of chaning basic parameters of the equation on the figure.Translations of the image and effects on the equation.Rotation of the image and effects on the equation.

**The initial example
presented is the variation of the ellipse.**

**An Ellipse and its
Reflection**

**As can be seen, these
pair of ellipses, provide the basis for our heart shape. From
the equation, however, some of the curve is unnecessary. This
is done by keeping the positive values of x unchanged, and reflecting
the negative values of x across the plane.**

**Also, by changing
the values for A, B, C and the constant F (presently all 1), we
can "shape" the ellipse.**

**Why This Works...**

**As can be seen from
the 4 graphs below, by changing the values of the coefficients
and the constant on the right-hand side of the equation, the shape
is modified.**

**The first example
(black and red) changes that constant. (Coefficient F from the
standard equation for conic shapes). As can be seen, as we increase
the value of the constant, the size of the shape increase accordingly.
In the graph shown, the values start at 100 and increase by 100
each step.**

**The second example
(blue) shows the effect of changing the Coefficient A value. As
the value of A increases, the ellipse starts to become more "rounded".
The lower the value for A, then the ellipse will elongate. As
from initial definitions for the ellipse, this coeeficient has
an affect on the horizontal (axis) size of the figure. The figure
is streched more along the x-axis than the y-axis**

**The third example
(red) shows the effect of changing the Coefficient B value. As
the value of B increases, the ellipse becomes more rounded as
B increases. The lower the value for B, then the ellipse rotates
and elongates along a different axis than with A. This is more
in line with the i mpact on the y-axis, than the x-axis as seen
with A.**

**The fourth example
(green) shows the effect of changing the Coefficient C value.
As the value of C increases, the ellipse becomes more elongated.
(almost an opposite effect from A). As C decreases, it begins
to fall into the more rounded shape of the ellipse.**

**Examples of Changing
Equation Parameters**

**If QuickTime is available,
use this link to see all A, B, and C coefficient changes dynamically:**

**Changing the Constant
Value:**

**Changing the First
Coeeficient:**

**Changing the Second
Coefficient:**

**Changing the Third
Coefficient:**

**Translations of the
Equation (Example at the beginning)**

The following shows a an example of translation of the same equation across the x-axis and the y-axis.

The basic shape fits into a area approximately 5 by 5 units. As a result, from simple tranposition of replacing x by x + 5 or x - 5 (and multiples of 5), we can translate the object either left or right, respectively. By changing y by y + 5 or y - 5 (and multiples of 5), we can also transpose the figure down or up respectively.

**Rotation of the Equation:**

The following shows a an example of rotation of the same equation about the origin.

Rotation is done by substituting: