tweaks

quarto/differentiable_vector_calculus/plots_plotting.qmd

@@ -373,7 +373,7 @@ surface(xs, ys, zs)
 :::{.callout-note}
 ## Note
 The above may not work with all backends for `Plots`, even if those that support 3D graphics.
-    :::
+:::
 
 For convenience, the `plot_parametric` function from `CalculusWithJulia` can produce these plots using interval notation, `a..b`,  and a function:
 
@@ -384,6 +384,31 @@ F(theta, phi) = [X(1, theta, phi), Y(1, theta, phi), Z(1, theta, phi)]
 plot_parametric(0..pi, 0..pi/2, F)
 ```
 
+##### Example, the general cone
+
+The general equation of cone consists of a vertex, $V$, and a base curve. The cone consists of all line segments connecting $V$ to the base curve, here parameterized and in the $x-y$ plane: $r(u) = \langle x(u), y(u), 0 \rangle$. The equations for the cone are:
+
+$$
+\frac{x - V_x}{x(u)-V_x} = \frac{y - V_y}{y(u)-V_y} = \frac{z - V_z}{z(u)-V_z} = t,
+$$
+
+where $t \in [0,1]$ This gives a vector-valued function
+
+$$
+F(u, t) = t * (r(u) - V) + V, \quad \alpha \leq u \leq \beta, 0 \leq t \leq 1.
+$$
+
+To illustrate, we have:
+
+```{julia}
+# cf. https://discourse.julialang.org/t/general-plotting-code-for-cone-in-3d-with-glmakie-or-plots/92104/3
+
+basecurve(u) = [cos(u), sin(u) + sin(u/2), 0]
+Vertex = [0, 3/4, 3]
+Cone(u, t) = t * (basecurve(u) - Vertex) + Vertex
+plot_parametric(0..2pi, 0..1, Cone)
+```
+
 ### Plotting  F(x,y, z) = c
 
 

quarto/differentiable_vector_calculus/vector_valued_functions.qmd

@@ -232,6 +232,7 @@ plot_parametric(0..2pi, f, legend=false)
 scatter!([0],[0], markersize=4)
 ```
 
+
 ##### Example