Bezier curve approximations

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Bezier curve approximations can be used to create shapes and figures.

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1 Bezier 1

2 Bezier 2

3 Bezier 3

4 Bezier 4

5 Bezier 6

In many software systems, Bezier curves are used to approximate circles since the human eye cannot tell the difference.

True circles and smooth curves, like points, lines, etc., in the mathematical sense, do not exist in the real world.

In computer graphics, curves, circles, etc., are approximated to sufficiently model what is desired.

Here is one quadrant of the curve.

Here are the control points with dotted lines for the entire circle.

Here is the drawn circle with the control points and dotted lines.

Here is the circle approximated with Bezier curves.

Can you see that this is not really a circle?

Can you tell the difference between the previous Bezier curve approximation of a circle and the following circle?

Note: It is possible that the PostScript software actually uses Bezier curves to draw the circle, in which case they would be exactly the same - neither one a real circle.

How many grooves are there in a LP record?

(Young people may not know what that means)

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1 Outline

2 Dots

3 Both

4 Lines

5 Fill

How would we create a vase shape image?

Start with some control dots. (grouped by 4 dots)

Here are the associated (dotted) lines. (grouped by 4 dots)

Now use Bezier curves to sufficiently approximate the desired curves. In artistic graphical design fields, designers specify control points using a mouse (or other pointing device) and the curves are filled in by the software.

Now fill in the shape with the desired color.

Software such as Pov-Ray can be used to add texture and reflections for realism.

Isaiah 64:8 But now, O LORD, thou art our father; we are the clay, and thou our potter; and we all are the work of thy hand. [kjv]

… πηλος … χειρων … [lxx]

Isaiah 64:8

All

KJV: But now, O LORD, thou art our father; we are the clay, and thou our potter; and we all are the work of thy hand.

Hebrew: ועתה יהוה אבינו אתה אנחנו החמר ואתה יצרנו ומעשה ידך כלנו׃

Greek: και νυν κυριε πατηρ ημων συ ημεις δε πηλος εργον των χειρων σου παντες

Isaiah 29:16

All

KJV: Surely your turning of things upside down shall be esteemed as the potter's clay: for shall the work say of him that made it, He made me not? or shall the thing framed say of him that framed it, He had no understanding?

Hebrew: הפככם אם כחמר היצר יחשב כי יאמר מעשה לעשהו לא עשני ויצר אמר ליוצרו לא הבין׃

Greek: ουχ ως ο πηλος του κεραμεως λογισθησεσθε μη ερει το πλασμα τω πλασαντι ου συ με επλασας η το ποιημα τω ποιησαντι ου συνετως με εποιησας

With mathematical approximations, the curves appear smooth.

Fractals provide roughness, but a "regular" roughness.

To be "natural" looking, there needs to be some roughness as apposed to smoothness.

And some randomness is needed. What exactly is meant by "random"? Koch snowflake