The talk is based on a teachware project of mathematics for science and engineering students of the first year. Teachware means a teaching aid to be open with a web browser. This presentation includes animated SVG graphics from very basic calculus and therefore it aims also the general audience without any mathematical orientation.

The **first part** of the talk is meant to be
short. An open source java program uses the
quadratic Bezier-Casteljau method for convenient expression of
two dimensional curves. A finite curve provided by the user is
approximated in number of intervals as quadratic expressions. The
number of intervals is automatically obtained by the required error
limit. The algorithm will be described and examples will be shown.
More details are available on the web
site of the program.

The **second part** will be the core of the
presentation with animated graphics. Three of them are a must and
if time permits, more can be added. Since they are written with the
purpose of teaching, each one of them consists of two parts
presented in parallel and modified consecutively: text with the
written explanation about the next stage and the graphics changing
appearance and animated.

The detailed list of the three animated presentations follows.

**Inversion of coordinates**-
What kind of transformation is necessary in order to invert a planar rectangular coordinate system? The

*first animation*shows that a simple rotation in two dimensions is not enough. The*second animation*shows a three dimensional rotation around a diagonal axis (of the plane) that perform the required inversion. In order to achieve the visual effect, SVG uses two static rotations and an animated scale transformation applied simultaneously. The*third animation*shows a two-dimensional mirror transformation that also solves the problem of inversion. The effect is achieved in SVG by animated stroke-opacity. **Measurement of the derivative**-
A cubic function with a minimum, a maximum and three roots serves as basis for the presentation. The tangent line of the curve is animated by use of auto-rotated animateMotion. On the next stage a none-rotated grid is added to the animated tangent that allows actual measurement of its slope, which is equivalent to the value of the derivative, at any point of the curve. Finally the analytically calculated derivative is plotted and compared to the measurements performed in the previous stage.

**Cycloid**-
The cycloid is a curve obtained by the motion of a point at the circumference of a rolling wheel without sliding along a straight line. An animation shows the motion by use of simultaneously applied animated rotation and translation. On the next stage the actual cycloid, obtained from the software described in the first part of this talk, is superimposed. Finally the wheel's motion is stopped and the analytical form of the curve is derived.