How would you express ##z## as a function of ##t##?Eve Litman said:Let's say you have a helix defined parametrically as
r(t) =<sin(t), cos(t), t>
Is it possible to eliminate t and write an equation for this helix just in terms of x, y, and z?
Thank you!mathman said:x=sin(z), y=cos(z).
The helix equation is a mathematical representation of a helix, which is a 3-dimensional shape that resembles a spiral. It is often used in scientific research to model various structures, such as DNA molecules, protein structures, and astronomical objects like galaxies.
Eliminating the parameter from the helix equation allows us to express the helix in terms of Cartesian coordinates (x, y, z) rather than in terms of a parameter (t). This makes it easier to analyze and manipulate the helix equation and extract important information from it.
The process involves solving the helix equation for the parameter (t) in terms of the Cartesian coordinates (x, y, z). This can be done by using trigonometric functions and basic algebraic manipulations. The resulting equation will be in the form of x = f(y,z) or y = f(x,z) or z = f(x,y), depending on which coordinate is chosen as the independent variable.
Eliminating the parameter allows us to easily plot the helix in 3-dimensional space and calculate important properties such as the radius, pitch, and number of turns. It also makes it easier to compare different helices and analyze their similarities and differences.
The process of eliminating the parameter may not always be possible for all types of helix equations. In some cases, it may result in a complex or undefined equation, making it difficult to extract meaningful information. Additionally, eliminating the parameter may also lead to loss of information, as the resulting equation may not fully capture the properties of the original helix equation.