Polar curve constant theta
Categories: polar coordinates
Here is a polar function of constant $\theta$.
$$\theta = 1$$
This function describes a curve where $\theta$ is constant for any value of $r$. In this case the angle is always 1 radian. The curve is the locus all points that are any distance $r$ from the origin in the direction of $\theta$. The curve in a half line, like this:
Notice that the curve is a half line, because in polar coordinates, $r$ cannot be negative.
This animation shows how the curve is plotted as $r$ increases from 0:
Varying the angle
In general the function:
$$\theta = a$$
where $a$ is a constant between 0 and $2\pi$, describes a half line at an angle $a$, starring at the origin.
Here are the lines created for $n = 1$ (red), $n = 2$ (green), and $n = 4$ (blue):
Cartesian equation
We can convert the polar equation into a Cartesian equation using the identity:
$$\tan{\theta} = \frac {y}{x}$$
Rearranging this equation gives:
$$y = x \tan{\theta}$$
which is the Cartesian equation for a straight line of gradient $\tan \theta$, passing through the origin. However, the equation in only valid for the quadrant in which $r$ would be positive.
See also
Join the GraphicMaths Newletter
Sign up using this form to receive an email when new content is added:
Popular tags
adder adjacency matrix alu and gate angle area argand diagram binary maths cartesian equation chain rule chord circle cofactor combinations complex modulus complex polygon complex power complex root cosh cosine cosine rule cpu cube decagon demorgans law derivative determinant diagonal directrix dodecagon eigenvalue eigenvector ellipse equilateral triangle euler eulers formula exponent exponential exterior angle first principles flip-flop focus gabriels horn gradient graph hendecagon heptagon hexagon horizontal hyperbola hyperbolic function hyperbolic functions infinity integration by parts integration by substitution interior angle inverse hyperbolic function inverse matrix irrational irregular polygon isosceles trapezium isosceles triangle kite koch curve l system line integral locus maclaurin series major axis matrix matrix algebra mean minor axis nand gate newton raphson method nonagon nor gate normal normal distribution not gate octagon or gate parabola parallelogram parametric equation pentagon perimeter permutations polar coordinates polynomial power probability probability distribution product rule proof pythagoras proof quadrilateral radians radius rectangle regular polygon rhombus root sech set set-reset flip-flop sine sine rule sinh sloping lines solving equations solving triangles square standard curves standard deviation star polygon statistics straight line graphs surface of revolution symmetry tangent tanh transformation transformations trapezium triangle turtle graphics variance vertical volume volume of revolution xnor gate xor gate