This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${\left\{P_i\right\}}_{i=1}^{\infty }$ generated by a three-term recurrence relation $P_i\left(x\right)+Q_1\left(x\right)P_{i-1}\left(x\right)+Q_2\left(x\right)P_{i-2}\left(x\right)=0$ with the standard initial conditions $P_0\left(x\right)=1,P_{-1}\left(x\right)=0,$ where $Q_1\left(x\right)$ and $Q_2\left(x\right)$ are arbitrary real polynomials.

We give a complete characterization of those $f:[0,1]\to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1,\mathrm{BV}}$ parametrization (i.e., a $C^1$ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for $X={ℝ}^n$. We present examples which show applicability of our characterizations. For example, we show that the $C^{1,\mathrm{BV}}$ and $C^2$ parametrization problems are equivalent for $X=ℝ$ but are not equivalent for $X={ℝ}^2$.