Explain how you sketch a plane curve given by parametric equations? What is meant by the orientation of the curve?
When sketching a plane curve given by parametric equations, you can follow these steps:
1. Identify the range of the parameter: Look at the given equations and determine the range of the parameter values that will generate a complete curve.
2. Find a few key points: Substitute different values of the parameter within the given range to obtain corresponding points on the curve. Choose values that cover various regions of the parameter range.
3. Plot the points: Once you have a few points, plot them on a coordinate system.
4. Connect the points: Trace a smooth curve that passes through the plotted points, following the general shape of the curve.
5. Consider symmetry: If the parametric equations have any symmetries (e.g., symmetry with respect to the x-axis or y-axis), utilize them to sketch the curve more accurately.
The orientation of a curve refers to the direction in which the curve is traced as the parameter varies. It indicates the order in which the points on the curve are connected.
To determine the orientation, you can examine the sign of the derivative of one parameter with respect to the other parameter. If the derivative is positive, the curve is traced in one direction (e.g., left to right or bottom to top), and if the derivative is negative, the curve is traced in the opposite direction. The orientation is important, especially when dealing with closed curves, as it affects the inside versus outside of the curve.