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Bezier curve approximations
1. Bezier curve approximations
Bezier curve approximations can be used to create shapes and figures.
2. Circle using Bezier curves
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.
3. Circles
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.
4. Control points
Here are the control points with dotted lines for the entire circle.
5. Draw the circle
Here is the drawn circle with the control points and dotted lines.
6. The circle
Here is the circle approximated with Bezier curves.
Can you see that this is not really a circle?
7. The real 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.
8. Grooves in a record
How many grooves are there in a LP record?
(Young people may not know what that means)
9. Vase using Bezier curves
1 Outline
2 Dots
3 Both
4 Lines
5 Fill
10. Vase
How would we create a vase shape image?
11. Dots
Start with some control dots. (grouped by 4 dots)
12. Dotted lines
Here are the associated (dotted) lines. (grouped by 4 dots)
13. Bezier curves
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.
14. Fill in
Now fill in the shape with the desired color.
Software such as Pov-Ray can be used to add texture and reflections for realism.
15. Isaiah 64:8
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]
και νυν κυριε πατηρ ημων συ ημεις δε πηλος εργον των χειρων σου παντες [lxx]
16. Isaiah 64:8
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
17. Isaiah 29:16
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
18. Smoothness
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"?
19. End of page