1. Introduction to Curves
UNIT V
Introduction to Curves

2. Introduction
Objects in the real world may not always be made up of regular geometric shapes.
Surfaces are made up of curved surfaces & curved edges.
Curves are quite complicated to represent them in exact mathematical equations.
Curves are classified into two categories- namable, unnamable.
Namable curves known as true-curve generation approach(planes, spheres, parabolas, circles, straight lines)

3. Curve continuity
To guarantee a smooth transition from one section of the piecewise curve to the next, we can enforce continuity conditions at the link points.
There are two types of curve continuities: geometric & parametric.
In geometric continuity, requires parametric derivatives of two curve sections to be proportional to each other at their common boundary instead of equal to each other.
Parametric continuity is obtained by matching the parametric derivatives of the adjoining two curve segments at their common boundary.

4. Conic curve
Curves are one of the most important primitive shapes to create high resolution graphics.
While using many small polylines allows to create graphic that appears smooth at fixed resolution and they do not preserve smoothness when scaled.
Curves can be stored much easier, can be scaled to any resolution without loosing smoothness and provide much easier way to specify real world objects.
A conic section or just conic is a plane curve that can be formed by intersecting a cone with a plane that does not go through the vertex of the cone.
Five types of conics are: Circle, Hyperbola, Parabola, Rectangular Hyperbola and Ellipse.
A circle and an ellipse are created when a cone and plane are intersected as a closed curve.

6. Spline curve representation
A spline is used to refer to a wide class of functions that are used in applications requiring data interpolation & smoothing.
These type of curves in the field of computer science because of the simplicity of their construction, accuracy of evaluation & their capacity to approximate complex shapes through curve fitting & interactive curve design.
It produces smooth curves through a designated set of points using piecewise cubic polynomial functions.
Two categories comes under this,
Interpolated Spline
Approximated Spline

7. Interpolation - the curve passes through all of the control points
Approximation - the curve does not pass through all of the control points

8. Bezier curve
Bezier curves must have precisely n control points, where n+1 is the degree of the Bezier polynomial.
By connecting bezier curves we can create long curves.
This is done by connecting last control point of one curve the same as first control point of next curve.
This type of curve always lie within the convex hull of the control points & always have the sum of the basis functions add to 1.
A convex hull of set of points is the smallest convex polygon that contains one of the points.

9. Bezier curve

10. B-spline curves fractal curves
A spline curve is a sequence of curve segments that are connected together to form a single continuous curve.
It refers to a spline curve parameterized by spline functions that are expressed as linear combinations of splines.
A B-spline is simply a generalization of a bezier curve.
fractal curves
A fractal is defined as a rough or fragmented geometric shape that can be split into parts, each of which is approx. a reduced-size reproduction of the complete shape, based on the property known as self similarity.
Many natural objects that approximate fractals to a degree, include clouds, mountains, coastlines.

11. B-spline curve
Applications of fractals

12. Surface generation
The surface plays a critical role in designing & manufacturing of automobile bodies, ship hulls, aircraft, games, furniture etc.
A surface in a 3D space is an object definition that has breadth, width which is specified with the help of equation.
Following are some types of surfaces:
Bilinear Surfaces
Ruled surfaces
Developable surfaces
Coons patch.

13. Bilinear surfaces: They are the simplest of all.
These types of surfaces are defined by means of their 4 corner points.
Their 4 boundary curves are straight lines & the coordinates of any point on this surface are derived by linear interpolations.

14. B) Ruled surfaces:
It is a surface swept out by a straight line as it moves through space.
Which means original figure stays fixed at one point but changes direction.

15. C) Developable surfaces:
These surfaces are characterized by only blending in one direction at a time like a cylinder or a cone.
Developable surfaces are useful because they allow round forms to be made out of a flat materials like plywood, sheet metal, or cloth etc.
D) Coons patch:
There are many other surfaces that needs to be deal within Computer-aided Geometric Design(CAGD), such as car bodies & airplane.
In these situations, a common approach is to divide the surface into patches defined by a network of curves.
Each patch is then parameterized by filling in or interpolating the given parameterizations of their boundary.

17. Painters Algorithm
The idea behind painters algorithm is to paint the polygon into frame buffer in order of decreasing distance.
The algorithm gets its name from the manner in which an oil painting is created. The artist begins with the background.
He then adds the most distant object and then the newer object and so on.
There is no need to erase the portion of background, the artist simple paints on top of them.
Whole canvas is painted first then on top this canvas a circle is painted
On top this circle a rectangle is painted.
On top of this rectangle an ellipse is painted.

18. It gives an appearance as the ellipse is nearer to the viewer then the circle.
The main factor here is to check the priority
Each polygon is assigned a priority number
Polygons are sorted according to their priority.
Lowest one first, followed by higher then highest.

19. Z-buffer algorithm
Z Buffer is one of the way to handle hidden lines and surfaces.
Here we are storing polygon which are closer to the viewer in the frame buffer.
Along with frame buffer we make use of Z Buffer which is a large array to store z coordinates(depth) of pixel which we want to display.
Depth buffer compares the two depth and chooses the depth nearer to the viewer
Used for detecting visible surfaces, comparing surface depths at each pixel position on the projection plane.
Also known as Depth buffering.

20. Advantages
It is easy to implement.
As Z buffer processes one object at a time total number of objects can be large.
It does not require any additional technique.
Disadvantages
It requires lot of memory as we are storing each pixel’s z value.
It is time consuming process as we are comparing each and every pixel.