I am trying to draw a clock with both hour and minute hands in a computer program. The movement of the clock hands would mirror a traditional wall clock (hours from $12, 1, 2, 3,..., 11$ and back to $12$ etc.) Suppose I know the center (starting point) and the radii of both the hour and the minute hands. In order to locate the endpoints (tip) of the clock hands, what parametric equation I should use? The complication seems to be that the hour hands start from $12$ (or we could assume it's "$0$") at the top of the clock. So, the traditional parametric equation for finding a point around a circle's circumference (see below) needs to be modified a bit here. If I get the hour hand equation down, I think I can adapt it for minute hand. My trig knowledge isn't that good, so if someone could provide me a solution (or even just a clue), that'd be very helpful. Thank you!
The conventional parametric equations of a circle are:
\begin{align} x &= r \cos(t)\\ y &= r \sin(t) \end{align} where $r$ is the radius and $t$ is in radian for above equations.
I found a document that derives how to graph the hour and minute hands, but the equations there (pasted below) doesn't seem to draw at least the hour hand right. For example, from $12$ to $3$ hour, it draws points in the fourth quadrant of a circle...
\begin{align} x &= r \cos(\frac\pi{360}t)\\ y &= r \sin(\frac\pi{360}t) \end{align} where $t$ is time in minutes.
This is NOT a proper answer to the request on parametric equations. However, if only the locations of the tips of the hands are of interest, you can consider the following:-
Let the lengths of the minute and hour hands be $M$ and $H$ respectively.
After 60 minutes = 1 hr, the minute hand moved $2 \pi M(1)$.
After 30 minutes = 0.5 hr, the minute hand moved $2 \pi M(0.5)$
After x minutes; (where $x \le 60$) $= \frac {x}{60}$ hour, the minute hand moved $2 \pi M(\frac {x}{60})$.
After 60 minutes = 1 hour, the hour hand moved $2 \pi H (\frac {1}{12})$.
After y hours (where $y \le 12$) $= \frac {y}{12}$ parts of a 12 hour revolution, the hour hand moved $2 \pi H(\frac {y}{12})$.