Find the length of the curve - is the astroid a smooth

Question

1. Find the length of the curve given by

r(t) = (cos t) + ln(tan(1/2t)),sin t)       Π/4 ≤ t ≤ 3Π/4

Hint: Since

sin(α + β) = sinα cosβ + cosα sinβ

we have that

sin t = 2 sin (1/2.t)cos(-2t).

2.  Let {p_a, p₁,......, p_k} be (k + 1) points in Rⁿ, where k ≤ n.

For all 1 ≤ i ≤ k define the vector v_i^→ by

v_i^→ = P_i - p_o

and let

V = span{v₁^→, v₂^→, ........ v_k^→}

The affine plane spanned by the points p₀,........., p_k, denoted A(P₀, .........p_k), is then defined as

A(po,......,pk) := po + V

= {q ∈ Rⁿ | q = p_o + v^→ for some v^→ ∈ V}.

Prove that q ∈A(p_o,......p_k) if and only if there exist constants α₀, α₁, ......., α_k

such that

q = _i=0Σ^k α_ip_i, and _i=0Σ^k α_i = 1.

3. The astroid curve in R² is given by the parameterization

γ(t) = (cos³ (t), sin³ (t) t ∈ [0, 2Π].

Is the astroid a smooth curve? Is it regular? Is it a curve of general type? If the answer is "no" for a property, characterize those arcs (or segments) of the astroid which have the given property. Lastly, show that the segment of any tangent line lying between the x and y axes has the same length (of 1).

4. Let -γ : [a, b] → R³ be a curve of general type. Suppose that the normal planes of γ all pass through a fixed point p. Prove that the curve lies on a sphere centered at p.