Multimedia: Vector Graphics Encoding

Introduction

Vector graphics are a way of representing images using mathematical equations that define lines, curves, and shapes. Unlike raster graphics (like JPEGs or PNGs) which store pixel data, vector graphics store instructions on how to draw the image. This makes them scalable without loss of quality.

Encoding Data for Vector Graphics

The data for a vector graphic is typically encoded as a series of commands and parameters. These commands instruct a graphics rendering engine on how to draw the image. The most common vector graphic formats include SVG (Scalable Vector Graphics), EPS (Encapsulated PostScript), and AI (Adobe Illustrator). While the specific encoding details vary between formats, the underlying principle is the same.

Common Vector Graphic Commands

Vector graphics formats use a set of commands to define the graphical elements. Some of the most common commands include:

• Line To (L): Draws a straight line from the current point to a specified point.
• Curve To (C): Draws a cubic Bézier curve from the current point to a specified point, using two control points to define the curve's shape.
• Quadratic Curve To (Q): Draws a quadratic Bézier curve from the current point to a specified point, using one control point to define the curve's shape.
• Circle To (C): Draws a circle with a specified center and radius.
• Rectangle To (R): Draws a rectangle with specified corner coordinates.
• Polygon To (P): Draws a closed polygon defined by a list of coordinates.
• Fill (F): Specifies the fill color for a shape.
• Stroke (S): Specifies the stroke color and thickness for a shape.

Data Structures for Vector Graphics

The encoded data for a vector graphic is often organized into a structured format. A common approach involves using a series of objects, where each object represents a distinct graphical element. Each object contains a sequence of commands and their associated parameters.

Example: SVG Encoding (Simplified)

SVG (Scalable Vector Graphics) is a popular XML-based vector graphics format. A simplified example of how a circle might be encoded in SVG is shown below:

Command	Parameters
circle	cx="50": x-coordinate of the circle's center cy="50": y-coordinate of the circle's center r="20": radius of the circle

This snippet indicates that a circle element is being defined with a center at (50, 50) and a radius of 20.

Mathematical Representation of Curves

Many vector graphics rely on mathematical curves, particularly Bézier curves, to create smooth shapes. Bézier curves are defined by a start point, an end point, and one or more control points. The control points influence the shape of the curve. The mathematical equation for a cubic Bézier curve is:

$$ B(t) = (1-t)^3 P_0 + 3(1-t)^2 t P_1 + 3(1-t) t^2 P_2 + t^3 P_3 $$ where:

• $B(t)$ is the point on the curve at parameter $t$ (where $0 \le t \le 1$)
• $P_0$, $P_1$, $P_2$, and $P_3$ are the coordinates of the control points and the start/end points

The parameters $P_0$, $P_1$, $P_2$, and $P_3$ define the shape of the curve, and the parameter $t$ determines the position along the curve.

Conclusion

In summary, vector graphics data is encoded using a set of commands and parameters that instruct a rendering engine on how to draw the image. These commands can define lines, curves, shapes, and their properties like color and thickness. Mathematical concepts, particularly those related to curves like Bézier curves, are fundamental to the creation of smooth and scalable vector graphics.