For k = 2

The two support points are

(x₁₁; x₁₂) = (cos θ₁₁ ; sinθ₁₁) and (x₂₁ ; x₂₂) = (cosθ₂₁ ;sinθ₂₁),

where .

The model matrix:

The two possible solutions are (θ₁₁, θ₂₁) = (1.4942, 0.0766) and (π, -π) with corresponding P_eff value, 0.5733 and 0.5714. The D-optimal added supports are (1.4101, 0.1082) and (0.1082, 1.4101). In fact, we can show that the |X^’X| are the same when (θ₁₁, θ₂₁) = (1.4942, 0.0766).

The response surface of |X^’X|

When (θ₁₁, θ₂₁) = (π, -π), the corresponding added points are (, 0) and (0, -). Basically, the |X^’X| are the same when we choose any two of four star points to be our added points.

Best Angle sampler

According to the above results, we believe that we can get the same resolutions by using different initial sets for BA sampler.

Example 1:

We assume T(t) = (5t)^-2/3 and N_t = 10. After 400 iterations of BA sampler, the optimal added points are

,

the corresponding P_eff value is 0.5714.

Example 2:

With the same assumption of T(t) and N_t, we use a set of different initial angles for the BA sampler. After 600 iterations of BA sampler, the optimal added points are

.

The P_eff value is 0.5733.