Special Functions
We will assume that our GP lets us plot multiple curves on the xy-plane by defining y(x) = \; ... \; for each. We will also assume that it supports \mathtt{pow, sqrt, log} as well as the basic the trigonometric functions \mathtt{cos}, \mathtt{sin}, \mathtt{tan}, \mathtt{acos}, \mathtt{asin}, \mathtt{atan} out of the box. Some less standard functions include \mathtt{abs, sgn, max, min}. If our GP doesn't support these then we can define them in the following manner.
\begin{array}{lll} \mathtt{abs}(a) & = \sqrt(a^2) \\ \mathtt{sgn}(a) & = \frac{a}{|a|} \\ \mathtt{max}(a,b) &= \frac{1}{2} \left( a + b + |a-b| \right) \\ \mathtt{min}(a,b) &= \frac{1}{2} \left( a + b - |a-b| \right) \\ \end{array}
Curve Segments
Suppose we want to plot only a segment of a curve, that is we want to restrict \gamma to the domain [a,b]. If our GP lets us plot parametric equations then this is a simple matter. We define
\begin{array}{llll} x(t) = t \\ y(t) = \gamma(t) \\ t \in [a,b] \\ \end{array}
Suppose however, that our GP only lets us plot y as a function of x. How can we define y(x) := \, ... so that it only plots \gamma on the interval x \in [a,b]? Let's take a look at a concrete example. We have the two curves \gamma: x \mapsto x+2 and \sigma: x \mapsto x(2x+1). Now we only want to plot the segment of \gamma where \gamma(x) \geq \sigma(x), ie. the interval [-1,1]. Here's how we can do it in GoodGrapher.
On the left we see the line \gamma in blue in its full glory. On the right we see \gamma restricted to the domain [-1,1]. The trick to plotting the segment is done by adding and subtracting \mathtt{asin}(x). GoodGrapher fails to evaluate \gamma outside [-1,1] because \mathtt{asin}(x) is undefined there. Generalizing, we let \Omega_{I} be the set of all real-valued functions that have a (maximal) domain I \subseteq \mathbb{R}^3. This means that if \omega \in \Omega_{I} then \omega(x) is well-defined (real and bounded) \mathtt{iff} x \in I. Combining functions by addition or composition yields the following rules.
\begin{align} \omega \quad & \in \quad \Omega_{I} \\ \phi \quad & \in \quad \Omega_{J} \\ \omega + \phi \quad & \in \quad \Omega_{\, I \, \cap \, J} \\ \omega \circ \phi \quad & \in \quad \Omega_{\, \{x \, \in\, J \; | \; \phi(x) \, \in \, I\}} \end{align}
Some examples include
\begin{align} \mathtt{asin}(x), \; \mathtt{acos}(x) \quad & \in \quad \Omega_{\, [-1,1]} \\ \; \sqrt{x+1}\, \sqrt{-x+1} \quad & \in \quad \Omega_{\, [-1,1]} \\ \sqrt{x} \quad & \in \quad \Omega_{\, [0,\infty)} \\ \mathtt{asin} \left( \tfrac{1}{b-a} (2x - a - b) \right) \quad & \in \quad \Omega_{\, [a,b]} \\ \mathtt{asin} \left( 2 \,\mathtt{cos}\, \pi x - 1 \right) \quad & \in \quad \Omega_{\, [0,1] \, \cup \, [2,3] \, \cup \, [4,5] \, \cup \, ...} \\ \mathtt{asin} (x^2 + y^2) \quad & \in \quad \Omega_{\,\mathbb{B}_2(1)} \end{align}
Notice how the composition-rule was used to define the function in \Omega_{\, [a,b]}. If we want to plot the segment x \in [a,b] of the curve \gamma we take any function \omega \in \Omega_{[a,b]} and then simply define y(x) := \gamma(x) + \omega(x) - \omega(x).
Curve Reflections
For example, reflecting about the line y=0 sends (x,y) \mapsto (x,-y). Reflecting about the line y=x sends (x,y) \mapsto (y,x). Reflecting about y = 1-x sends (x,y) \mapsto (1-y,1-x). Plotting the reflected curve is straightforward in parametric form.
Curve Rotations
Combining Regions
We will represent a region F in the xy-plane canonically by f(x,y) < 0, where we may replace the < with \leq to include the boundary. For example, the inside of the circle centered at the origin with radius 1 can be defined by x^2 + y^2 - 1 < 0. While GoodGrapher let's you shade such regions, it does not include a way to take the intersection or union. However, there is an easy workaround.
\begin{align} A \quad & : \quad a(x,y) < 0 \\ B \quad & : \quad b(x,y) < 0 \\ A \cap B \quad & : \quad \mathtt{max}(a,b) < 0 \\ A \cup B \quad & : \quad \mathtt{min}(a,b) < 0 \\ A \cup B - A \cap B \quad & : \quad ab < 0 \end{align}
\begin{array}{llll} x(t) = t \\ y(t) = \gamma(t) \\ t \in [a,b] \\ \end{array}
Suppose however, that our GP only lets us plot y as a function of x. How can we define y(x) := \, ... so that it only plots \gamma on the interval x \in [a,b]? Let's take a look at a concrete example. We have the two curves \gamma: x \mapsto x+2 and \sigma: x \mapsto x(2x+1). Now we only want to plot the segment of \gamma where \gamma(x) \geq \sigma(x), ie. the interval [-1,1]. Here's how we can do it in GoodGrapher.
On the left we see the line \gamma in blue in its full glory. On the right we see \gamma restricted to the domain [-1,1]. The trick to plotting the segment is done by adding and subtracting \mathtt{asin}(x). GoodGrapher fails to evaluate \gamma outside [-1,1] because \mathtt{asin}(x) is undefined there. Generalizing, we let \Omega_{I} be the set of all real-valued functions that have a (maximal) domain I \subseteq \mathbb{R}^3. This means that if \omega \in \Omega_{I} then \omega(x) is well-defined (real and bounded) \mathtt{iff} x \in I. Combining functions by addition or composition yields the following rules.
\begin{align} \omega \quad & \in \quad \Omega_{I} \\ \phi \quad & \in \quad \Omega_{J} \\ \omega + \phi \quad & \in \quad \Omega_{\, I \, \cap \, J} \\ \omega \circ \phi \quad & \in \quad \Omega_{\, \{x \, \in\, J \; | \; \phi(x) \, \in \, I\}} \end{align}
Some examples include
\begin{align} \mathtt{asin}(x), \; \mathtt{acos}(x) \quad & \in \quad \Omega_{\, [-1,1]} \\ \; \sqrt{x+1}\, \sqrt{-x+1} \quad & \in \quad \Omega_{\, [-1,1]} \\ \sqrt{x} \quad & \in \quad \Omega_{\, [0,\infty)} \\ \mathtt{asin} \left( \tfrac{1}{b-a} (2x - a - b) \right) \quad & \in \quad \Omega_{\, [a,b]} \\ \mathtt{asin} \left( 2 \,\mathtt{cos}\, \pi x - 1 \right) \quad & \in \quad \Omega_{\, [0,1] \, \cup \, [2,3] \, \cup \, [4,5] \, \cup \, ...} \\ \mathtt{asin} (x^2 + y^2) \quad & \in \quad \Omega_{\,\mathbb{B}_2(1)} \end{align}
Notice how the composition-rule was used to define the function in \Omega_{\, [a,b]}. If we want to plot the segment x \in [a,b] of the curve \gamma we take any function \omega \in \Omega_{[a,b]} and then simply define y(x) := \gamma(x) + \omega(x) - \omega(x).
Curve Reflections
Suppose we want to plot a curve reflected about a given line. Let our reflection line be y = ax + b. Upon reflection, the point (x,y) gets mapped to
\begin{align} (x,y) \quad & \longmapsto \quad (\,\chi , \, \gamma\,) \\ \\ \chi(x,y) \quad & = \quad \tfrac{1}{a^2+1} \,[\,(1-a^2)x + 2ay -2ab\,] \\ \gamma(x,y) \quad & = \quad \tfrac{1}{a^2+1}\, [\,2ax + (a^2-1)y + 2b\,] \end{align}
\begin{align} (x,y) \quad & \longmapsto \quad (\,\chi , \, \gamma\,) \\ \\ \chi(x,y) \quad & = \quad \tfrac{1}{a^2+1} \,[\,(1-a^2)x + 2ay -2ab\,] \\ \gamma(x,y) \quad & = \quad \tfrac{1}{a^2+1}\, [\,2ax + (a^2-1)y + 2b\,] \end{align}
Curve Rotations
Suppose we want to rotate a curve about a given pivot (p_x, p_y). Rotating the point (x,y) counter-clockwise by an angle \theta equates to the mapping.
\begin{align} x \quad & \mapsto \quad p_x \; + \; (x-p_x) \cos \theta \; - \; (y-p_y) \sin \theta \\ y \quad & \mapsto \quad p_y \; + \; (x-p_x) \sin \theta \; + \; (y-p_y) \cos \theta \end{align}
\begin{align} x \quad & \mapsto \quad p_x \; + \; (x-p_x) \cos \theta \; - \; (y-p_y) \sin \theta \\ y \quad & \mapsto \quad p_y \; + \; (x-p_x) \sin \theta \; + \; (y-p_y) \cos \theta \end{align}
Combining Regions
We will represent a region F in the xy-plane canonically by f(x,y) < 0, where we may replace the < with \leq to include the boundary. For example, the inside of the circle centered at the origin with radius 1 can be defined by x^2 + y^2 - 1 < 0. While GoodGrapher let's you shade such regions, it does not include a way to take the intersection or union. However, there is an easy workaround.
\begin{align} A \quad & : \quad a(x,y) < 0 \\ B \quad & : \quad b(x,y) < 0 \\ A \cap B \quad & : \quad \mathtt{max}(a,b) < 0 \\ A \cup B \quad & : \quad \mathtt{min}(a,b) < 0 \\ A \cup B - A \cap B \quad & : \quad ab < 0 \end{align}
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